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IN  MEMORIAM 
FLORIAN  CAJORl 


RATIONAL   ARITHMETIC, 

IN    WHICH    THE 

SCIENCE  IS  FULLY  DEVELOPED, 

THE 

ART    C  L  E  A  R  T.  Y    T^  X"  P  I    A  I  \  E  D , 

AND   BOTH   COMBINED    IN    NUMEROUS    ILLUSTRATIONS  ; 
ADAPTED   TO   LEABNER4 

OF  EVERY  CAPACITY. 

THL    WHOLE    ENFORCED   BY    A   GREAT   VARIETY   OF 

INTERESTING    AND   PRACTICAL    PROBLEMS. 

TO    WHICH    IS    APPENDED,  * 

Jk  EISY, 

CONTAINING    THE    ANSWERS   TO   THE    PROBLEMS. 


BY   J.   S.  RUSSELL, 

TXACHKR  OP  MATHBMATICS  kit  TRB  LOWBLL  HIOB  SCHOOL. 


SECOND      E4)ITI0N. 


LO  WELL: 

PUBLISHED  BY  THOMAS  BILLINGS 

BOSTON      B.  B.  MUSSEY. 

1S47 


Entered  according  to  Act  of  Congress,  in  the  yesr  1846,  by 

J.  S.  RUSSELL, 
In  the  Clerk's  Office  of  the  District  Court  of  Massachusetu. 


<>    Stereotyped   bj 
GEORGE    A.    CURTIS; 
MBW   MH^LAMV  TITS   AXC  •TBAKOTT7S  VOUWMIV 


w 


PREFACE. 


Every  public  school  may  be  divided,  in  respect  to  the  stad^  of 
Arithmetic,  into  three  classes.  The  first  and  smallest  class  either 
possess  by  nature,  or  have  happily  acquired,  a  taste,  and,  conse- 
quently, a  talent  for  the  study.  For  them  there  is  no  imperious 
necessity  of  adding  another  to  the  numerous  treatises  already  m  use ; 
for,  although  they  will  meet  with  much  difficulty,  through  mdefinite 
and  confused  modes  of  expression  and  incomplete  demonstrations,  in 
arriving  at  the  philosophy  of  the  subject,  yet,  in  spite  of  these  obsta- 
cles, they  will  eventually  comprehend  the  important  principles  of 
Arithmetic,  and,  what  is  remarkable,  adopt  the  same  modes  of 
expression  which  have  so  much  opposed  their  own  progroBS. 

Tq  this  class  the  Rational  Arithmetic,  though  not  indispensable, 
will  be  of  very  essential  service.  Had  all  learners  been  of  this  class, 
however,  the  author  would  have  been  spared  the  labor  and  expense 
he  has  devoted  to  this  work. 

But  there  is  a  class  of  medium  ability,  including  about  one  half, 
who  may  be  saved  incalculable  labor  and  vexation,  by  using  this 
book,  while  pursuing  this  difficult  study. 

It  is  expected,  however,  that  the  third  class  will  most  truly  appre- 
ciate this  work.  They  include  about  one  third,  and  consist  of  those 
unfortunate  scholars  whose  minds  act  too  slowly  for  the  patience  of 
teachers,  and  are  too  obtuse  to  derive  much  advantage  from  the  text- 
books so  ill-adapted  to  their  wants. 

It  is  for  these  two  classes,  the  last  in  particular,  that  the  Rational 
Arithmetic  is  prepared ;  to  their  wants  it  is  thought  to  be  well 
adapted  ;  and  it  is  expected,  that  they  will  hereafter  assume  a  more 
just  relative  standing  with  their  schoolmates ;  not  waiting,  as  hereto- 
fore, for  the  results  of  business  life  ib  prove  them  possessed  of  minds, 
less  active  indeed,  yet  not  inferior  in  strength  and  capacity  of  im- 
provement. 

The  author  knows  no  other  written  Arithmetic  that  is  adapted  to 
learners ;  they  seem  to  him  rather  books  of  reference  for  those  who 
already  understand  the  subject,  and  are  able  to  perceive  the  princi- 


IV  PREFACE. 

pies  Without  exp.anation.  Some,  indeed,  have  attempted  to  meet 
the  wants  of  learners,  by  introducing  the  several  subjects  by  a  page 
or  two  of  puerile  questions  which  arc  seldom  noticed  eitherby 
teachers  or  scholars.  Others  found  the  principles  upon  the  imagi- 
nary answers  to  be  ffiven  to  such  questions  by  those  acknowledged 
to  be  ignorant.  Such  must  be  a  very  uncertain  foundatioiif 
especially  so  when,  as  sometimes  happens,  theee  l^-^'^-^"  v^estioiie 
are  so  misformed  as  to  lead  astray.     Instanci*  the  :    *•  In 

11,  how  much  more  does  the  1  in  the  lens'  place  8tiv,.„  .  .  :aan  the 
1  in  the  unit's  place?  In  880,  how  much  more  does  the  8  in 
the  hundreds'  place  stand  for  than  the  8  in  the  tens'  place  1  It  is 
just  so  in  all  cases ;  therefore,  A  figure  at  the  left  of  another  stands 
for  ten  times  as  much  as  it  would  m  the  place  of  that  other  figure.*^ 
The  simple  learner  will,  probably,  understand  these  questions  literally, 
and  give  for  answers,  so  far  as  he  is  able,  9  more,  720  more,  &c., 
between  which  and  the  principle  purporting  to  be  derived  from 
them,  there  is  no  direct  connection.     Had  these  questions  been  thus, 

How  many  times  as  much as 1  insiea'*  "  .w 

much  more than 1  ^  they  would  have  i  1- 

ligent  mind  directly  to  the  principle.  In  presence  of  the  u;^nur  to 
correct  false  answers,  to  sum  up  and  enforce  the  cooclueion,  such 
questions  properly  asked  are  well  enough  ;  but  otherwise,  they  are 
extremely  vexatious  and  discouraging,  and  meet  echolars  will  pass 
over  them  without  adding  to  their  knowledge. 

In  the  Rational  Aritlnnetic  it  has  been  the  object  to  prepare  matter 
for  the  intelligent  study  of  the  learner  by  himself  y  that  in  due  time  he 
may,  in  a  well  conducted  recitation,  exhibit  with  credit  and  pleasure, 
both  to  himself  and  teacher,  his  thorough  knowledge  of  the  lesson. 
Such  results  are  far  different  from  ordinary  experience.  Teachers 
who  have  desired  to  ground  their  pupils  in  the  principles  of  the 
science,  while  the  text-books  have  failed  to  afford  the  necessary 
instruction,  have,  by  oral  instruction,  and  black-board  iUustrations, 
endeavored,  with  only  partial  success,  to  effect  this  object,  at  present, 
so  indispensable.  Such  instruction,  though  efficient  with  tlie  more 
inteUigent  and  active  minds,  proves  insufficient  for  a  large  portion  of 
every  school.  It  is  expected  that  the  Rational  Arithmetic  will  come 
to  the  relief  of  such  teachers,  enabling  them  with  less  labor  to  secure 
much  happier  results. 

The  peculiarities  of  the  Rational  Arithmetic  are  :  1st,  A  philosoph- 
ical arrangement y  and  systematic  treatment  of  the  several  subjects. 
Multiplication  and  Division,  being  only  particular  cases  of  Addition 
and  Subtraction,  respectively,  follow  their  heads  in  the  natural  order, 
in  the  fundamental  principles ;  but  in  fractions  and  compound  num- 
bers, from  the  greater  convenience,  they  resume  the  common  order 
again.  T\\e  subjects  under  the  head  of  Percentage,  being  applica- 
tions of  the  principle  of  Proportion,  very  properly  follow  Proportion. 
The  study  of  Arithmetic  being  now  so  extensive,  it  is  no  longer 
necessary  to  place  Interest  nearer  the  beginning  of  the  book  than  its 


PREFACE.  T 

proper  place,  to  insure  a  knowledge  of  it.  Indeed,  all  the  subjecta 
are  so  arranged  that  each  is  explained  on'  principles  previously 
taught. 

2d.  A  complete  devehpmeiU  of  the  fundamental  principles.  Nume- 
ration, in  particular,  both  integral  and  fractional,  the  foundation  of 
the  whole  superstnirture,  has  received  especial  attention. 

3d.  A  full  (!■  L  and  thorough  explanation  of  the  various 

applications  of  i  nontal  principles. 

4th.  A  <  jard  to  the  abridgment  of  labor,  by  viewing  num- 

bers ihro!  liictors,  and  relations,  canceling  common  factors 

when  both  m.  ••-i  and  division  are  involved  in  the  same  pro- 

cess ;  and  alwa  ig-'upon  fractions  in  a  manner  to  secure  the 

simplest  terms  i ...^. 

5th.  The  giving  of  a  reason  for  everything  stated,  and  in  such 
style  that  the  repetition  of  the  language  will  induce  in  the  learners 
the  understanding  of  the  reason  which  it  embodies. 

6th.  Numerous  references  to  other  parts  of  the  book  for  informa- 
tion bearing  upon  the  subject  in  hand. 

7th.  An  exclusion  of  all  such  indefinite  expressions  as  **  5  times 
greater,"  **  seven  times  too  large,"  "seven  times  too  small,"  **  in- 
creases it  ten  times,"  **  5  times  too  great,"  "  100  times  larger  or 
smaller  ;"  and  all  such  provoking  expressions  as  **  it  is  obvious,"  "  it 
is  plain,"  **  evidently,"  &c.,  whose  office  is  only  to  occOpy  the 
place  of  an  inconvenient  reason.  These  peculiar  excellences,  it  b 
thought,  warrant  the  title  assumed  for  the  book. 

It  is  recommended  to  teachers,  although  the  author  knows  no  writ- 
ten arithmetic  so  easily  understood,  that  the  younger  pupils,  pre- 
viously to  taking  up  this  book,  shall  have  well  studied  Colbunrs 
First  Lessons,  or  some  other  intellectual  arithmetic ;  but  when  it 
is  taken  up,  that  they  accommodate  their  speed  to  thoroughness ; 
that  they  take  special  notice  of  the  numerous  references  to  other 
parts  of  the  book,  where  their  memories  may  be  refreshed  with 
necessary  information  upon  the  present  subject. 

Teachers  will,  of  course,  as  far  as  circumstances  admit,  classify 
their  pupils,  assign  lessons,  and  hear  recitations  in  this,  as  in  other 
studies.  Although  each  is  left  to  his  own  experience  and  tact  in 
conducting  recitations,  yet  we  may  urge  the  importance  of  securing 
in  some  way  a  thorough  analysis  of  everything^  the  giving  of  a  reason 
for  each  step  in  the  solution  of  problems,  showing  its  bearing  upon,  or 
tendency  to  war  Is  the  final  result.  In  the  author's  experience,  it  has 
proved  well  to  require  the  pupils  to  bring  to  the  recitation,  not  only 
the  results,  but  the  written  process  of  their  work ;  also  to  exhibit 
their  skill  in  the  solution  of  problents  upon  a  black-board  sufficiently 
ample  \  >  accommodate  the  whole  class.  The  problems  may  be 
those  of  the  ordinary  lesson,  or  such  as  may  be  suggested  on  the 
occasion  ;  and  the  recitation  should  be  such  as  shall  exhibit  the 
scholar's  knowledge  of  the  principles  involved  in  the  process. 

The  impossibility  of  preventing  the  access  of  the  scholars  to  th« 

1* 


¥1  PREFACE. 

"  Key  published  for  the  use  of  teachers  only^^*  the  immenae  injury 
done  to  the  moral  sense  by  the  futile  attempt,  the  securing  of 
greater  faithfulness  on  the  part  of  teachers,  and,  on  the  other  hand, 
the  accommodation  of  the  better  scholars,  who  dislike  to  have  the 
answers  obtruded  upon  their  notice  before  they  shall  have  given  the 
problem  a  fair  trial,  with  other  reasons,  have  induced  ih'^  author  to 
ap|)end  the  Key  to  the  Arithmetic,  where  it  may  be  cm  v, 

and  innocently  accessible  to  all.    Hut  should  any  prefer  li  <! 

^  ;ite,  by  signifying  their  desire  to  the  publisher,  ii  may  tx) 

i «)  111*;  numerous  friends  who  advised  tliis  undertaking,  who  have 
encouraged  its  progress,  and  arc  ready  to  receive  it,  and  give  it  **  a 
start  in  the  world,"  the  author  takes  this  occasion  to  express  his 
gratitude.  He  also  acknowledges  his  obligations  to  the  numerous 
authors  whose  works  he  has  consulted ;  yet  few  will  discover  any- 
thing heretofore  published,  except  a  few  select  problems,  as  the 
Rational  Arithmetic  is  chiefly  derived  from  an  experience  of  more 
than  ten  years'  teaching  of  the  mathematics,  half  of  which  has  1 
devoted  exclusively  to  arithmetic. 

Lowell,  Oaober   1846. 


TABLE    OF    CONTENTS. 


NUMERATION. 


Definitions,        

Fonnatien  of  Numbers,        .... 

Arabic  Figures, 

Expression  of  Numbers,      .... 
Absolute  and  Relative  Value  of  Units, 

Reading  of  Numbers, 

Writing  of  Numbers, 


ADDITION. 


Definitions,  and  Use  of  Signs,       .    .    • 

Addition  Table,       

Written  Process  and  Proof  of  Addition, 


MULTIPLICATION. 


V'  '■  "  "   -IS, •     .     .     . 

fion  Table,       

iuuiin'iyiiig  by  one  Digit,  .... 
Composite  and  Prime  Numbers,  .  . 
Factors  being  Abstract  Numbers, 

Multiplying  by  one  Unit  of  any  order, 

Multiplying  by  any  number  of  Units  of  the  same  order, 
Inverting  the  Order  of  the  Factors,    .... 

Proof  of  Multiplication, 

General  Explanation  of  Multiplication,  .  . 
Factors  expressing  Units  of  the  higher  orders, 
General  Exercises, 


SUBTRACTION. 


Subtraction  Illustrated,    .    .    . 

Definitions, 

Subtraction  Table,  .... 
Proof  of  Subtraction,  .  .  . 
Subtraction  "without  Reduction, 
Subtraction  requiring  Reduction, 


DIVISION. 


Division  Illustrated,  and  Definitions,      .     , 
Division  Table,       ....•..,, 
Dividend  expressing  Units  of  any  one  order. 
Division  requiring  Reduction,       .... 


■BCTIOn. 

1 

2 

3 

4  to  11 

12 

13  —  14 

15—16 


17  —  18 

19 

20  —  23 


24  —  25 
26 
27  —  28 
29  —  31 
32  —  33 
34  —  35 
36  —  37 
38  —  39 
40  —  41 
42  —  45 
46  —  48 
49 


50 
51 
52 

53  ^54 
55 

56  —  60 


61  —  64 

65 

66—67 

68  —  69 


CONTENTS. 


Writ>;cn  Process  and  Proof  of  Division, 

Short  Process  of  Division , 

Long  Process  of  Division 

General  Exercises,  


FRACTIONS 


Origin  and  Mode  of  writing  Fractioi 

Definitions, 

Reading  of  Fractions, 
Expression  of  Division, 
Finding  the  Whole  from  a  Tail,    . 
Finding  a  Part  from  the  Whole, 

]\I()des  of  considering  and  ^  "  ' ^  ;   <  ...n^. 

Expression,  Definitions,  :i  n  o|  Fraction.*-. 

Reduction  of  Mixed  Nuin  'T  Fractions,  . 

Multiplication  of  Fractioi  umbers,    .     . 

Division  of  Fractions  by  1 :  is,         .     ,     . 

Dividing  by  the  Factors  of  ihc  Divisor, 
Reduction  of  Fractions  to  lower  terms. 

'•     -  '■^'       -rs,         

H'lor, 

^ — V ,  ,...iiple  and  Denominator 

Addition  and  Subtraction  of  Fractions, 

Multiplying  by  Fractions, 

Dividing  by  Fractions, 

Review  of  Multiplication  of  Fractions  by  Fractions,    . 
Review  of  Division  of  Fractions  by  Fractions,    .     .     . 

DECIMAL  FRACTIONS. 
Similarity  of  Decimal  Fractions  and  Integral  Numbers, 
Local  Value  of  r»™'  »•''•""■'- 
Reading  of  Dt'  • 
Writing  of  lk\ 

Federal  Money  expressed  by  Decimal  Number- 
Reduction  of  Federal  Money, 

Addition  and  Subtraction  of  Decimals,        

Multiplication  of  Decimals, 

Reduction  of  Common  Fractions  to  Decimals,      .     .     . 

Dividing  by  Units  of  the  higher  orders, 

Infinite  Decimals,        

Division  of  Decimol- 

COMrOU.ND  NU3IBERS. 

Definitions,        

Tables, 

Reduction  of  Compound  Numbers,         

Addition  of  Compound  Numbers, 

Subtraction  of  Compound  Numbers, 

Multiplication  of  Compound  Numbers, 

Division  of  Compound  Numbers, 


■  BCnOH. 

70  to  76 

77  —  78 

79  —  80 

81 


82^85 
83 

84 
86to8S 
89  —  90 
91—9:; 


100—109 
110—116 
117—119 
120  —  130 
123—124 
128—130 
131  —  140 
141  —  143 
144  — 150 
151  — 157 
158  —  159 
160—161 


162 
163 
164  —  165 
166  —  168 
169 
170—172 
173  -  174 
175—177 
178—179 
180  —  182 
183  —  187 
188—191 


192 
193—206 
207—233 
234  —  236 
237  —  239 
240  —  24 
242  —  243 


CONTENTS.  9 

CBcnoif. 

Using  of  Numbers  variously  expressed, 244  lo  245 

Finding  the  Difference  of  Time  between  Dates,  ....  246  —  24B 

Reduction  of  Compound  Numbers  for  Multiplication,       .  248  —  249 

Mensuration  of  Surfaces  and  Solids,  250  —  251 

Abridged  Solutions  of  Problems,  252  —  253 

Reduction  of  Currencies, 254  —  255 

Practice,  or  the  Use  of  Aliquot  Parts, 256  —  263 

PROPORTION. 

Ratio,       264-265 

TVT'iitiniving  by  Ratios, 266  —  271 

g  by  Inverse  Ratios, 272  —  275 

] 276—278 

]  (jportion, 279  —  280 

c  Proportion, 281  —  284 

Conjoined  Proportion, 285  —  286 

Barter, 287  —  290 

Fellowship,        291  —  292 

Compound  Fellowship, 293  —  294 

PERCENTAGE. 

Percentage,         295  —  297 

Commission, 298  —  300 

Stocks, 301  —  302 

Insurance, 303  —  304 

Assessment  of  Taxes, 305 

Duties, 306  —  309 

Interest, 310  —  313 

Interest  on  Partial  Payments, 314  —  316 

Banking,        317  —  320 

ComJ)ound  Interest, 321 — 323 

Compound  Interest  on  Partial  Payments, 324  —  325 

Problem  to  find  the  Time, 326  —  328 

Problem  to  find  the  Per  Cent., 329  —  331 

Problem  to  find  the  Rate  Per  Cent., 332  —  334 

Problem  to  find  the  Principal, 335  —  337 

Problem  to  find  the  Present  Worth, 338  —  340 

Problem  to  find  the  Discount, 341  —  343 

Problem  to  find  the  Face  of  a  Discounted  Note,  .     .    .     .  344  —  346 

Problem  for  the  Equation  of  Payments, 347  —  349 

ALLIGATION. 

Problem  to  find  the  Average  of  Ingredients, 350  —  352 

Problem  to  find  the  Quantities  of  Ingredients,      ....  353  —  356 

Problem  to  mix  Ingredients  partially  limited,      ....  357  —  359 

Problem  to  mix  a  Limited  Compound,* 360  —  362 

POWERS  AND  ROOTS. 

Definitions  and  Illustrations, 363  —  364 

Extraction  of  the  Square  Root, 365  —  372 

Extraction  of  the  Cube  Root,        373  —  381 


10 


CONTENTS. 


SERIES. 


r  in  Scries  by  Difference, 

>  find  either  Extreme  and  the  Sum, 
)  find  the  Common  Difference  and  Sum, 
)  find  the  Number  of  Temw  and  Sum, 

i  iis  in  Series  by  Quotient, 

Problem  to  form  Series,        

Problem  to  find  either  Extreme  and  Po-wcr  of  the  Ratio 

Problem  to  find  the  Sum, 

Infinite  Series,        .     .     . 

Com;         '  'forest  by  Senes, 

Com  ount  by  Series^ 

Aniiu...; ..ned,       .... 

Annuities  at  Simple  Interest,    . 

Annuities  at  Compound  Interest 


MENSURATION. 

Definitions, 

Mcnsnmtion  of  the  Parallelogram  and  Triangle,     .     . 

!on  of  the  r 

ion  of  the  I 

ion  of  the  Cyiimier,         

ion  of  the  Pyramid,  Cone,  and  Wedge,      .     . 

ion  of  the  Frustum  of  the  Pyramid  and  Cone, 

on  of  the  Sphere, 

<       _    _  of  Casks, «... 


REVIEW. 

Review  of  Fractions, 

Review  of  Compound  Numbers,      .     . 
Review  of  Proportion, 
Review  of  Percentage, 

Review  of  Alligation, 

Review  of  Powers  and  Roots,     . 

Review  of  Series, 

Review  of  Mensuration, 

Curious  Problems, 


382 
383  to  385 
386  —  388 
389  —  391 

392 
393  —  396 
397  —  400 
401  —  403 
404  —  406 
407  —  409 
410  —  412 

413 
414  —  419 
420  —  428 


429 

430  —  433 
433  —  435 
436  —  437 
438  —  439 
440  —  441 
442  —  443 
444  —  445 
446—447 


448 
449 
450 
451 
452 
453 
454 
455 
456 


KEY. 
Containing  the  Answers  to  all  the  Problems. 


% 


ARITHMETIC. 


I.    NUMERATION. 

!•   Arithmetic  Defined. 

Arithmetic  is  the  science  of  numbers,  and  the  art  of  com- 
puting by  them. 

As  a  science,  arithmetic  explains  the  nature  and  properties 
of  numbers ;  and  demonstrates  the  principles  and  rules  for 
the  practice  of  the  art. 

As  an  art,  arithmetic  explains  the  methods  of  working  by 
numbers  for  the  solution  of  numerical  problems. 
3,   Formation  of  Numbers. 

A  single  thing  of  aoy  kind  is  called  a  unit,  or  One. 

The  larger  numbers  are  formed  by  the  successive  addition 
of  units.  Thus,' if  to  one,  another  unit  of  the  same  kind  be 
added,  the  collection  forms  the  number.  Two. 

The  collection  of  two  and  one  forms  the  number,     Three. 

The  collection  of  three  and  one  forms  the  number.  Four. 

The  collection  of  four  and  one  forms  the  number.     Five. 

The  collection  of  five  and  one  forms  the  number.      Six. 

The  collection  of  six  and  one  forms  the  number.       Seven 

The  collection  of  seven  and  one  forms  the  number.  Eight. 

The  collection  of  eight  and  one  forms  the  number.   Nine. 

In  like  manner,  the  addition  of  one  unit  to  any  number, 
forms  the  next  larger  number. 

3.    Arabic  Figures. 

« 

Among  the  \^arious  methods  of  expressing  numbers,  the 
Arabic  is  superior  ;  and  is  now  in  general  use.  According 
to  this  method,  all  numbers  can  •be  expressed  by  different 
combinations  of  one,  or  more,  of  ten  figures. 

The  figures  are,  1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

The  first  nine  figures  are  also  called  digits.  And  each 
digit,  expressing  one  of  the  first  nine  numbers,  has  the  samp 
name  as  the  number  which  it  expresses. 


12  ARITHMETIC. 

Thus:  One      is  expressed  by  this  figure,  1,  called  One. 
^  Two     is  expressed  by  this  figure,  2,  called  Two. 

J  Three  is  expressed  by  this  figure,  3,  called  Three. 

Four    is  expressed  by  this  figure,  4,  called  Four. 
Five     is  expressed  by  this  figure,  5,  called  Five. 
Six      is  expressed  by  this  figure,  6,  called  Six 
Seven  is  expressed  by  this  figure,  7,  called  ^ 
Eight  is  expressed  by  this  figure,  8,  called  L.^.... 
Nine    is  expressed  by  this  figure,  9,  called  Nine. 
The  other  figure,  0,  called  Cipher,  unlike  the  digits,  does 
not  express  a  number,  nor  have  any  value;  but  yet,  as  we 
shall  see,  it  is  not  without  its  use. 

4.  Expression  of  Tens,  or  Units  of  the  Second  Order. 

There  is  no  appropriate  figure  to  express  the  next  number, 
called  ten,  or  any  of  the  larger  numbers ;  but  these  same 
digits  are  made  to  express  other  numbers  by  occupying  dif- 
ferent places  in  relation  to  each  other.  ^^  hen  they  stand 
alone,  or  in  the  Jirst  place,  each  expresses  a  certain  number 
of  units  of  the  first  order.  But  ten  units  of  this  order  are 
considered  collectively  as  forming  one  unit  of  the  secoTid 
order  ;  and  the  digits  are  made  to  express  units  of  the  second 
order,  called  tens,  by  occupying  the  second  place  from  the 
right  hand.     Thus:  • 

10  is  one     ten,  called  Ten. 

20  is  two    tens,  called  Twenty. 

30  is  three  tens,  called  Thirty. 

40  is  four    tens,  called  Forty. 

50  is  five     te^is,  called  Fifty. 

60  is  six      te/is,  called  Sixty. 

70  is  seven  tens,  called  Seventy. 

SO  is  eight  tens,  called  Eighty. 

90  is  nine  te?is,  called  Ninety. 
Here  the  digits  express  ten  times- as  much,  or  numbers  ten 
limes  as  large,  as  when  they  stand  in  the  first  place,  or  alone, 
because  they  occupy  the  second  place,  and  not  because  there  is 
any  value  in  the  cipher.  The  cipher  merely  occupies  the 
fiist  place,  in  order  that  there  may  be  a  second  place  for  the 
dif^it  to  occupy.  So  always,  the  cipher  is  used  to  occupy 
places  where  nothing  of  value  is  needed ;  but  which  must  be 
occupied,  in  order  that  the  digits  required  for  the  expression 
of  the  number,  may  stand  in  tli^ir  proper  plsces. 


NUBCERATION.  13 

S*  Expression  OF  Numbers  from  Ten  to  One  Hundred. 

The  numbers  between  the  tens,  that  is,  between  ten  and 
twenty,  twenty  and  thirty,  &c.,  are  expressed  by  making 
every  digit,  in  succession,  occupy  the  first  place,  together 
with  each  digit  in  the  second  place.     Thus : 


s  one  unit  of  the  second  order,  called  Ten. 

s  ten  and  one,  called  Eleven. 

s  ten  and  two,  called  Twelve. 

s  ten  and  three,  culled  Thirteen. 

8  ten  and  four,  called  Fourteen. 

s  ten  and  five,  called  Fifteen. 

s  ten  and  six,  called  Sixteen. 

s  ten  and  seven,  called  Seventeen. 

s  ten  and  eight,  called  Eighteen. 

s  ten  and  nine,  called  Nineteen. 

8  two  tens,  or  two  units  of  the  second  order,  called  Twenty 

s  two  tens  and  one,  called  Twenty-one. 

s  two  tens  and  two,  called  Twenty-two. 

s  two  tens  and  three,  called  Twenty-three. 

s  two  tens  and  four,  called  Twenty-four. 

s  two  tens  and  five,  called  Twenty-five. 

s  two  tens  and  six,  called  Twenty-six. 

s  two  tens  and  seven,  called  Twenty-seven. 

s  two  tens  and  eight,  called  Twenty-eight. 

is  two  tens  and  nine,  called  Twenty-nine. 

s  thr^  tens,  or  units  of  the  second  order,  called  Thirty. 

s  three  tens  and  one,  called  Thirty-one. 


10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28^ 

29 

30 

31 

32  &c.,  is  three  tens  and  two,  called  Thirty-two. 

41  &c.,  is  four  tens  and  one,  called  Forty-one. 

51  &c.,  js  five  lens  and  one,  called  Fifty-one. 

61  &c.,  is  six  tens  and  one,  called  Sixty-one. 

71  &c.,  is  seven  tens  and  one,         called  Seventy-one. 

81  &c.,  is  eight  tens  and  one,  called  Eighty-one. 

91  &c.,  is  nine  tens  and  one,  called  Ninety-one. 

99  16  nine  tens  and  nine,  called  Ninety-nine. 

6.    Expression   of  Hundreds,   or  Units   of   the    Third 
Ord^r. 
Ninety-nine  is  the  largest  number  that  can  be  expressed 
by  two  figures.     The  next  larger  number  is  ten  tens,  or  ten 
units  of  the  second  order,  which  are  considered  collectively  as 
forming  one  unit  of  the  third  order,  called  one  hundred. 
2 


14  ARITHMETIC. 

This  unit  is  also  expressed  by  the  tirst  digit;  but,  it  being 
a  unit  of  the  third  order,  the  digit  is  put  in  the  third  place. 
And  the  other  digits,  by  occupying  the  third  place,  are  made 
to  express  units  of  the  third  oraer,  or  hundreds.     Thus : 

100  is  One  hundred.  600  is  Six  hundred. 

200  is  Two  hundred.  700  is  Seven  hundred. 

300  is  Three  hundred.  800  is  Eight  hundred. 

400  is  Four  hwidred.  900  is  Nine  hundred. 
500  is  Five  hundred. 

T.   Expression  op  Numbers  from  One  Hundred  to  Onb 
Thousand. 

The  numbers  between  the  hundreds  are  expressed,  by 
making  all  the  numbers  less  than  one  hundred,  in  succession, 
occupy  their  own  places  at  the  right  of  each  digit  in  the 
third  place.     Thus: 

101  is  one  hundred  and  one. 
210  is  two  hundred  and  ten. 
311  is  three  hundred  and  eleven. 
425  is  four  hundred  and  twenty-five. 
543  is  ^\e  hundred  and  forty-three. 
608  is  six  hundred  and  eight. 
717  is  seven  hundred  and  seventeen. 
876  is  eight  hundred  and  seventy-six. 
999  is  nine  hundred  and  ninety-nine. 

§•   Expression  of  Thousands,  or  Units  of  the  Fourth 
Order. 

Nine  hundred  and  ninety-nine  is  the  largest  number  that 
can  be  expressed  by  three  figures.  The  next  larger  number 
is  ten  hundreds,  or  ten  units  of  the  third  order,  which  are 
considered  collectively  as  forming  one  unit  of  the  fourth 
order,  called  07ie  thoitsand. 

To  express  thousands,  or  units  of  the  fourth  order^  the 
digits  are  put  in  \\ie  fourth  place, 

9.    Expression  of  Numbers  from  One  Thousand  to  Ten 

Thousand. 

Any  number  between  the  thousands  is  expressed  by  using 
such  digits  as  are  needed  in  their  proper  places  at  the  right 
of  the  thousands.     Thus  : 


NUMERATION.  ij 

1000  is  one  thousand. 

2001  is  two  thousand  and  one.  .     .'* 

3020  is  three  thousand  and  twenty. 

4500  is  four  thousand  and  five  hundred. 

5055  is  five  thousand  and  fifty-five. 

6107  is  six  thousand  one  hundred  and  seven. 

7819  is  seven  thousand  eight  hundred  and  ninetecii. 

SOU  is  eight  thousand  and  eleven. 

9999  is  nine  thousand  nine  hundred  and  ninety-nine. 

10.  Expression  of  Ten-Thousands  and  Units  of  other 

Orders. 

Ten  units  of  the  fourth  ardery  form  one  unit  of  the  fifth 
order,  called  a  ten-thoiLsand.  And  the  ten-thousands  must 
occupy  the  fifth  place. 

In  the  same  manner,  higher  orders  of  units  are  formed,  to 
an  unlimited  'extent ;  ten  units  of  any  order  forming  one  unit 
of  the  next  higher  order,  to  be  expressed  in  the  next  higher 
place  ;  while  the  lower  places  are  used  for  the  expression  of 
units  of  lower  orders. 

Whence  it  follows  that  one  unit  of  any  order  equals  ten  units 
of  the  next  lower  order;  this  law  prevailing,  even  below 
units  of  the  first  order,  to  an  unlimited  extent,  as  will  be 
shown,  (183«)  Hence  the  law  of  the  local  value  of  figures 
is,  that  any  digit,  by  each  removal  to  the  next  higher  place, 
is  made  to  express  ten  times  as  much,  and  by  each  removal 
to  the  next  lower  place,  one  tenth  as  much  as  it  would  before 
such  removal. 

N.  B.  The  term  unit  means  a  unit  of  the  first,  or  lowest 
order,  unless  otherwise  specified. 

11.  T^BLE  exhibiting  THE  FORMATION,  NaME,  AND  EXPRES- 
SION OF  ONE  Unit  of  each  of  the  first  Ten  Orders. 

A  single  thing  of  any  kind  forms  one  Unit,  1. 

Ten  units  of  the  same  kind  form  one  Ten,  10. 

Ten  tens  form  one  Hundred,  100. 

Ten  hundreds  form  one  Thousand,  1000. 

Ten  thousands  form*one  Ten-thousand,      10000. 

Ten  teu-thousands  form  one  Hundred-thous.  100000. 

Ten  hundred-thousands       form  one  Million,  1000000. 

Ten  millions  form  one  Ten-million,   10000000. 

Ten  ten-millions  form  one  Hundred-million,  100000000. 

Ten  hundred-millions  form  one  Billion.  1000000000. 


16  ARITHMETIC. 


o  S 

.    <D  •-     -     O    2     rt    p 

These  units  of  ten  different  orders  §  ^  J  §  ^  * c 

may  be  expressed  in  one  number.      ;^  —  ^;^^xb^r^Xb'^ 

Thus:  1111111111 

since  each  unit  now  occupies  the  place  appropria 
units  of  its  own  order.     This  number  is  read,  One  li 
one  hundred  and  eleven    Million,  one  hundred  and  eleven 
Thousand,  one  hundred  and  eleven. 

19*  Illustration  of  the  Absolute  and  Relative  value 
OF  THE  Unit  of  different  Orders. 

Suppose  there  should  be  a  country  having  ten  states,  each 
state  having  ten  cities,  each  city  having  ten  schools,  each 
school  having  ten  classes,  each  class  having  ten  scholars,  and 
each  scholar  having  ten  cents  to  pay  for  a  writing-book.  How 
many  cents  would  it  take  to  buy  a  book  for  a  scholar  ? — ^books 
for  a  class  ? — for  a  school  ? — for  a  city  ? — for  a  state  I — for 
the  country  ?     It  would  take 

for  one  scholar,  ten  times  1  ct.,    equal  to  10  cts. 

for  one  class,       ten  times  10  cts.,  equal  to  100  cts. 

for  one  school,  ten  times  100  cts.,  equal  to  1000  cts. 
for  one  city,  ten  times  1000  cts.,  equal  to  10000  cts. 
for  one  state,  ten  times  10000  cts.,  equal  to  100000  cts. 
for  the  country,  ten  times  100000  cts.,  equal  to  1000000  cts. 

Other  answers. — It  would  take,  one  cent  being  a  unit  of  the 
first  order, 

for  one  scholar,  a  Ten-cent-piece, 

which  is  a  unit  of  the  second  order ; 

—  for  one  class,  a  Dollar-bill, 

which  is  a  unit  of  the  third  order ; 

—  for  one  school,  a  Tcn-dollar-bill, 

which  is  a  unit  of  the  fourth  order ; 
— for  one  city,  a  Hundred-dollar-bill, 

which  is  a  unit  of  the  fifth  order ; 
— for  one  state,  a  Thousand-dollar -bill, 

which  is  a  unit  of  the  sixth  order ; 
— for  the  country,  a  Ten-thousand-dollar-bill, 

which  is  a  unit  of  the  seventh  order. 


NUMERATION.  17 

Suppose  one  man  should  make  all  these  writing-books,  and, 
having  done  them  up  in  one  package,  should  sell  it  to  the 
president  of  the  country  ;  of  this  package  the  president  should 
make  ten  equal  packages,  and  sell  one  of  them  to  the  gover- 
nor of  each  state  ;  each  governor  should  make  of  his  pack-^ 
age  ten  equal  packages,  and  sell  one  of  them  to  the  mayor  of 
each  city ;  each  mayor  should  make  of  his  package  ten  equal 
packages,  and  sell  one  of  them  to  the  teacher  of  each  school ; 
each  teacher  should  make  of  his  package  ten  equal  packages, 
and  sell  one  of  them  to  the  head  scholar  of  each  class,  and 
each  head  scholar,  on  opening  his  package,  should  find  just 
ten  books,  nine  of  which  he  should  sell  to  his  class,  and  keep 
the  other  himself. 

One  book  being  a  unit  of  the  first  order,  a  unit  of  what 
order  would  be  each  head  scholar*s  package  ? — each  teacher's 
package? — each  mayor's  package? — each  governor's  pack- 
age?—  the  president's  package  ? 

How  many  books  in  the  president's  package? — in  each 
governor's  package? — mayor's  package?  —  teacher's  pack- 
age ? — head  scholar's  package  ? 

Suppose  each  one  should  pay  the  cents  for  his  book,  or 
package,  to  the  person  from  whom  he  had  received  it.  Then 
each  scholar  would  pay  his  head  scholar  nearly  a  handful  of 
cents ;  each  head  scholar  would  pay  his  teacher  nearly  a 
**  double  handful ;"  each  teacher  would  pay  his  mayor  nearly 
three  pints ;  each  ma^or  would  pay  his  governor  nearly  two 
pecks ;  each  governor  would  pay  the  president  nearly  6ye 
bushels ;  and  the  president  would  pay  the  book-binder  nearly 
Mv  bushels,  and  just  a  million  of  cents. 

it  would  take  the  book-binder's  son  more  than  two  months 
to  count  them,  if,  instead  of  going  to  school,  he  should  count, 
at  the  rate  of  one  cent  every  second,  for  three  hours  every 
half  day,  except  Saturday  afternoons  and  Sundays. 

13.   Manner  of  reading  Numbers. 

In  any  number,  the  first  three  figures  express  so  many 
hundreds  tens  and  units  of  Uiiit^;  the  second  three,  so  many 
hundreds  tens  and  units  of  Thousands ;  the  third  three,  so 
many  hundreds  tens  and  units  of  Millions ;  and  each  suc- 
ceeding three,  so  many  hundreds  tens  and  units  of  Billions, 
Trillions^  Qitadrillioyis,  Quintillions,  Sextillions,  Septillions, 
Octillions,  NonillioTis,  DeciUionSt  &c.,  respectively. 
2* 


18 


ARITHMETIC. 


Hence,  to  read  numbers,  count  off  the  figures  from  the 
right,  into  periods  of  three  figures  each,  and  beginning  at  the 
left,  read  each  period  separately,  as  so  many  hundreds  tens 
and  units,  naming  each  period  as  it  is  read,  except  the  right 
hand  period,  which  is  understood  to  be  units  without  its 
name  being  called.     Thus  : 

ill!    1    I  .1,-1 

31  562  896  125  944  361  299. 

14.   Exercises  in  reading  Numbers 
Read  the  following  numbers, 

1,  19.  10,       20907. 

2,  70.  11,      417016. 

3,  98.  12,     5008S40. 

4,  502.  13,     40910008. 

5,  610.  14,    136000200. 

6,  847.  15,     6500004. 

7,  1005.  16,   1147865479. 

8,  5049.  17,      416000. 

9,  9153.  18,  900001317601. 

IS*   Manner  of  writing  Numbek 

To  tarite  numbers  in  figures,  first  write  the  left  hand 
period,  which  may  require  one,  two,  or  three  figures,  then, 
m  succession,  write  the  other  periods,  allowing  three  places 
for  each  period. 

Write  in  figures  nine  quintillion,  six  hundred  and  one 
quadrillion,  ten  trillion,  nine  hundred  eighty-two  billion,  five 
million,  and  six  hundred,  9,601,010,982,005,000,600. 

In  this  number,  9  only  must  occupy  the  period  of  quintil- 
lions,  601  the  period  of  quadrillions,  010  the  period  of  tril? 
lions,  982  the  period  of  billions,  005  the  period  of  millions, 
000  the  period  of  thousands,  and  600  the  period  of  units. 

16»    Exercises  in  writing  Numbers. 

In  like  manner  write  the  foUotoing  numbers, 

1.  One  hundred  and  three. 

2.  Three  hundred  and  one. 


h 

1 

1         1 

1 

111  1    i 

2-ti     Ig        J 

£5 

i 

I  if!    I      5 

9  601  010  982  005  000  600 

UMBERS 

19, 

iL^(ifn:^79oori<^074. 

20, 

!)0f;i79^(i:W)LHi. 

21, 

:n790000i9. 

22, 

K)OG0400700091. 

23, 

SI 1123365. 

24, 

347000016011. 

25, 

333311112222. 

26, 

89000066000044000. 

27, 

.55000000100200010. 

ADDITION.  19 

3.  One  thousand,  and  ten. 

4.  Two  thousand,  one  hundred  and  seven. 

5.  Twenty  thousand,  and  thirty. 

6.  Fifty  thousand,  seven  hundred  and  five. 

7.  Three  hundred  thousand,  and  fifty. 

8.  Seven  hundred  and  seven  thousand,  seven  hundred  and 
twenty. 

9.  One  million,  three  hundred  and  seventy. 

10.  Five  million,  six  hundred  thousand,  and  seventy-three. 

11.  Five  hundred  ninety  million,  forty-seven  thousand,  and 
eight. 

12.  Three  billion,  six  hundred  seventy  million,  three  hun- 
dred and  two. 

13.  Forty-five  billion,  seven  million,  seventy  thousand,  and 
seven. 

14.  Fifty  trillion,  six  hundred  fifty-seven  million,  and  five 
hundred. 

15.  Six  trillion,  se^'en  hundred  and  three  billion,  twenty 
million,  and  twelve. 

16.  Seventy-seven  million,  ten  thousand,  and  nineteen. 

17.  Eight  billion,  ^ve  hundred  and  thirty  thousand. 
19.  Forty-nine  billion,  three  hundred  and  sixty. 

19.  Eighty  six  quadrillion,  ten  billion,  one  hundred  mil- 
lion, and  sixty. 


II.    ADDITION. 

17,   Addition  Defined  and  Illustrated. 

Addition  is  the  uniting  of  two,  or  more,  numbers  to  form 
another  number  equal  to  their  sum.     Thus : 

1.  If  you  should  place  5  cents  in  a  pile,  and  on  that  pile 
put  3  more  cents ;  how  many  cents  would  there  be  in  the 
pile? 

You  are  taught,  (9j)  that  numbers  are  formed  by  succes- 
sive additions  of  one  unit ;  but  here  you  are  required  to  form 
a  number  by  the  addition  of  3  urfits.  This  can  be  done  by 
adding  the  3  units,  1  at  a  time  ;  thus,  5  cents  and  1  cent  are 
6  cents,  and  1  cent  are  7  cents,  and  1  cent  are  8  cents,  which 
is  the  whole  number  of  cents  in  the  pile. 

2.  A  man  paid  a  5  dollar-bill  for  a  pair  of  boots,   and  a 


20 


ARITHMETIC. 


3  dollar-bill  for  a  pair  of  shoes ;  how  many  dollars  did  he 
pay  away  ?  ^ 

To  answer  this  question,  you  must  add  3  dollars  to  5  dol- 
lars as  we  added  3  cents  to  5  cents  in  the  1st  example.  But 
if  you  remember  what  the  sum  of  5  and  3  is,  you  need  not 
add  the  3,  one  at  a  time,  but  all  at  once,  saying  5  dollars  and 
3  dollars  are  8  dollars ;  therefore,  he  paid  away  8  dollars. 

18.  Explanation  and  Use  of  Signs. 

For  convenience  and  brevity,  signs  are  often  employed  in 
arithmetic.  Thus :  =  Tivo  horizontal  lines  are  the  sign  for 
eqimlity.  It  implies  that  what  precedes  the  sign  equals  what 
follows  it ;  as  100  cents  =  1  dollar ;  read,  100  cents  equal  1 
dollar. 

-(-  The  right  cross  is  the  sign  for  addition.  It  implies  that 
the  number  which  follows  the  sign  is  to  be  added  to  what 
precedes  it,  as  5-1-3  =  8;  read  5  plus  3  equal  S. 

Plus  is  the  Latm  word  for  more,  and  means  here  the  same 
as  if  you  should  say  5  more  3,  or  5  and  3  more  equal  8. 

19.  Addition  Table. 

In  order  to  perform  addition  with  facility,  you  will, 
before  attempting  further  progress,  correctly  ascertain,  and 
thoroughly  commit  to  memory,  the  sum  of  each  combination 
of  two  numbers  in  the  following  table. 


2  &  2  are 

3  &  2,  or  2  &  3  are 

4  &  2,  or  2  &  4  are 

5  &  2,  or  2  &  5  are 
()  &  2,  or  2  &  6  are 

7  &  2,  or  2  &  7  are 

8  &  2,  or  2  &  8  are 

9  &  2,  or  2  &  9  are 

3  &  3  are 

4  <5c  3^  or  3  &  4  are 

5  &  3,  or  3  &  5  are 

6  &  3,  or  3  6c  6  are 


7  &  3,  or  3  &  7  are 

8  &  3,  or  3  &  S  are 

9  &  3,  or  3  &  9  are 

4  &  4  are 

5  &  4,  or  4  &  5  are 

6  &  4,  or  4  &  6  are 

7  &;  4,  or  4  &  7  are 

8  &  4,  or  4  &  8  are 

9  &  4,  or  4  &  9  are 

5  &:  5  are 

6  &  5,  or  5  &  6  are 

7  &  5,  or  5  &  7  are 


8  &  5,  or  5  &  8  are 

9  &  5,  or  5  &  9  are 

6  &  6  are 

7  &;  6,  or  6  &  7  are 

8  &  6,  or  6  &;  8  are 

9  &  6,  or  6  &  9  are 

7  &  7  are 

8  &  7,  or  7  &  8  are 

9  &  7,  or  7  &  9  are 

8  &  8  are 

9  &  8,  or  8  &  9  are 
9  &  9  are 


90.    Explanation  of  the  Written  Process  of  AoDrrioN. 

1.    A  man  paid  25  dollars  for  a  cow,  and  3  dollars  for  a 
sheep ;  how  many  dollars  did  they  cost  ? 


ADDITION.  21 

Under  the  25  write  the  3,  so  that  it  shall  stand  in  a  column 

with  the  5 ;  and,  since  both  the  5  and  3  express  units 

25         of  i\ie  first  order ^  (4,)  add  them  together,  and  write 

3         8,  their  sum,  directly  under  them,  in  the  first  place, 

23         and, the  2  expressing  units  of  the  second  order,  write 

it  beside  the  8,  in  the  second  place,  which  gives  28 

dollars  for  the  answer. 

2.  A  man  having  sold  the  produce  of  his  farm,  received  48 
dollars  for  potatoes,  25  dollars  for  wheat,  32  dollars  for  rye, 
28  dollars  for  corn,  and  54  dollars  for  hay  ;  how  many  dollars 
did  he  receive  ? 

Arrange  the  numbers  together,  so  that  the  units  of  each 
order  shall  stand  in  a  column ;  then  ascertain  the 
48  sum  in  the  lowest  column ;  thus,  8  and  5  are  13, 
25  and  2  are  15,  and  8  are  23,  and  4  are  27  units  of  the 
32  first  order  ;  but  since  ten  units  of  any  order  make  one 
28  unit  of  the  next  higher  order,  (lOj)  these  27  units 
54         of  the  first  order  will  make  2  units  of  the  second 

order  and  7  units  of  the  first  order ;  hence*  write  the  7 

187         in  the  first  place,  and  add  the  2  with  those  of  the  same 

kind  in  the  second  column  ;  thus,  2  and  4  are  6,  and 
2  are  8,  and  3  are  11,  and  2  are  13,  and  5  are  18  units  of  the 
second  order,  making  8  units  of  the  second  order,  and  1  unit 
of  the  third  order;  therefore,  write  the  8  in  the  second 
place,  and  the  1  in  the  third  place,  which  gives  187  dollars 
for  the  answer. 
81  •   Proof  of  Addition. 

To  prove  the  correctness  of  any  operation  in  addition,  repeat 
the  operation,  combining  the  figures  of  each  column  in  the  op- 
posite  order.    If  the  two  results  agree,  probably  both  are  correct. 

22.   Model  of  a  Recitation. 

What  is  the  sum  of  the  following  numbers,  35468,  503. 
2300,  95  and  90072? 

Arrange  these  numbers  together,  so  that  the  units  of  each 

order  shall  stand  in  a  column ;    2   and  5  are  7, 

35468         and  3  are  10,  and  8  are  18  units,  equal  to  8  units, 

503         which  write  in  the  units*  place,  and  1  ten,  which 

2300         add  with  the  tens ;    1  and  7  are  8,  and  9  are  17, 

95         and  6  are  23  tens,  equal  to  3  tens,  which  write 

90072         in  the  tens'  place,  and  2  hundreds,  which  add 

with  the  other  hundreds  ;   2  and  3  are  5,  and  5 

128438         are   10,  and  4  are  14  hundreds,  equal  to  4  hun- 


22  ARITHMETIC. 

dreds,  which  write,  and  1  thousand,  which  add  with  the  other 
thousands ;  1  and  2  are  3,  and  5  are  8  thousands,  which 
write  ;  9  and  3  are  12  ten-thousands  ^  equal  to  2  ten -thou- 
sands, which  write,  and  1  hundred-thousand,  which  write, 
giving  128438,  the  sum  required. 

HenA:e,  observe  ;  that  in  addition,  the  units  of  each  order ^ 
he  ginning  with  the  loioest,  are  added  separately,  and  reduced  ^ 
(2O85)  as  far  as  may  be,  to,  and  added  laith  units  of  the 
next  higher  order,  writing  in  each  place  only  the  excess  over 
exact  units  of  the  next  higher  denomination, 

^23.   Exercises  in  Addition. 

In  like  manner,  solve  and  explain  the  following  proble?ns. 

1.  Mr.  Sampson  sold  6  loads  of  potatoes,  measuring, 
severally,  36,  34,  38,  28,  29,  and  33  bushels;  how  many 
bushels  did  he  sell  ? 

2.  Mr.  Mason  bought  5  hogs,  weighing,  severally,  375, 
358,  416,  410,  and  400  pounds ;  how  many  pounds  did  all 
weigh  ? 

3.  Mr.  Thomson's  wagon  weighed  2097  pounds,  and  the 
load  of  hay  on  the  wagon,  1988  pounds  ;  what  was  the 
weight  of  both  ? 

4.  Mr.  Wilson  sold  6  fat  oxen,  weighing,  severally,  907, 
1216,  1189,  1075,  899,  and  934  pounds ;  what  was  the 
weight  of  all  ? 

5.  How  many  strokes  does  a  clock,  which  strikes  the 
hours,  strike  in  12  hours. 

6.  How  many  days  in  a  year,  there  being  in  January  31 
days ;  in  February  28  ;  in  March  31  ;  in  April  30 ;  in  May 
31 ;  in  June  30 ;  in  July  31 ;  in  August  31 ;  in  September 
30 ;  in  October  31 ;  in  November  30 ;  and  in  December  31  ? 

7.  Mr.  Johnson  bought  a  farm  with  the  buildings  and 
stock  upon  it,  paying  5000  dollars  for  the  land,  2500  for  the 
house,  975  for  the  barn,  507  for  the  other  buildings,  and 
1650  for  the  stock  and  farming  tools;  how  many  dollars  did 
all  these  things  cost  him  ? 

8.  Mr.  Jackson  paid  4150  dollars  for  land,  2000  dollars 
for  a  house,  725  dollars  for  a  barn,  609  dollars  for  other 
buildings,  and  1200  dollars  for  stock  and  tools ;  what  was 
the  whole  cost  ? 

9.  Mr.  Jameson  paid  10000  dollars  for  a  factory,  5967 
for  land,  8096  for  cotton,  4870  for  labor,  and  908  for  team- 
ing ;  how  many  dollars  do  these  sums  amount  to  ? 


ADDITION.  23 

[lat  is  the  sum  of  all  the  numbers  that  you  speak 
in  counting  one  hundred  ? 

'  11.  How  many  square  miles  in  the  New  England  States, 
there  being  in  Maine  35000  ;  in  New  Hampshire  9491 ;  in 
Vermont  8000 ;  in  Massachusetts  7800 ;  in  Rhode  Island 
1225 ;  and  in  Connecticut  4764  ? 

12.  How  many  square  miles  in  the  Middle  States,  there 
being  in  New  York  46085 ;  in  New  Jersey  8320 ;  in  Pennsyl- 
vania 47000  ;  and  in  Delaware  2100  ? 

13.  How  many  square  miles  in  the  Southern  States,  there 
being  in  Maryland  9356  ;  in  Virginia  70000  ;  in  North  Caro- 
lina 50000 ;  in  South  Carolina  33000  ;  in  Georgia  62000  ; 
in  Alabama  51770 ;  in  Mississippi  48000 ;  and  in  Louisiana 
48320? 

14.  How  many  square  miles  in  the  Western  States,  there 
being  in  Tennessee  45000;  in  Kentucky  40000;  in  Ohio 
44000 ;  in  Indiana  36400 ;  in  Illinois  55000 ;  in  Michigan 
60000 ;  in  Missouri  64000 ;  and  in  Arkansas  55000  ? 

15.  How  many  square  miles  in  the  26  states,  mentioned 
in  the  last  four  problems  ? 

16.  If  the  time  from  the  creation  of  the  world  to  the 
deluge  was  1656  years,  thence  to  the  building  of  Solomon's 
temple  1344  years,  thence  to  the  birth  of  Christ,  1004  years ; 
how  old  is  the  world  in  the  year  of  our  Lord  1846  ? 

17.  How  long  since  the  deluge  ? 

18.  How  old  was  the  world  at  the  birth  of  Christ  ? 

19.  How  long  since  the  building  of  Rome,  which  was  753 
years  before  Christ  ? 

20.  How  long  since  Lycurgus  established  his  laws  at 
Lacedaemon,  which  was  131  years  before  the  building  of 
Rome  ? 

21.  How  many  miles  from  Augusta  in  Maine,  to  New 
Orleans  in  Louisiana,  it  being  from  Augusta  to  Portland  53 
miles,  thence  to  Boston  118  miles,  to  Hartford  160,  to  New 
York  123,  to  Philadelphia  90,  to  Baltimore  100,  to  Wash- 
ington  38,  to  Richmond  123,  to  Raleigh  165,  to  Charleston 
265,  to  Savannah  113,  to  Talahassee  331,  to  Mobile  320,  and 
to  New  Orleans  160  ? 

22.  How  far  from  Natches  in  Mississippi,  to  Boston  in 
Massachusetts,  it  being  to  Tuscaloosa  350  miles,  to  Nash- 
ville 230,  to  Louisville  210,  to  Cincinnati  110,  to  Wheeling 
230,  to  Pittsburg  115,  to  Buffalo  160,  to  Albany  360.  and  to 
Boston  150? 


ARITHMETIC. 


III.    MULTIPLICATION. 

S4.   Multiplication  Defined  and  Illustrated. 

Multiplication  is  the  producing  of  a  number  equal  to  as 
many  times  one  given  number  as  there  are  units  in  another 
given  number.     Thus : 

In  1  bushel  are  32  quarts.  How  many  quarts  in  8  bushels  ? 
Since  there  are  32  quarts  in  one  bushel,  in  8  bushels  there 
are  8  times  32  quarts,  the  amount  of  which  may  be  ascer- 
tained by  addition.  But  when  the  amount  of  several  times 
the  same  number  is  to  be  ascertained,  it  can  be  done  by  a 
shorter  process.     Instead  of  writing  the  32  quarts  8  times,  as 

1st  Operation.     2d  Operation.  ^^     the      Ist     OperatioU,     witC    it    Ouly 

once,  as  in  the  2d  operation,  and 
under  it  write  8,  to  show  how  many 
times  the  32  should  be  taken.  Then 
j.^g  8  times  2  units,  (which  is  the  same  in 
amount  as  the  eight  2's  of  the  units' 
column  in  the  1st  operation,)  are  16 
units,  equal  ( 10)  to  6  units,  which 
write  in  the  units'  place,  and  1  ten, 
which  reserve  to  join  with  the  other 

256  quarts,  ^^"^-  .  ^  ^'"^^^  ^  ^^"^;  (^^^^^i?  ^^^ 

same  m  amount  as  the  eight  J  s  of 

the  tens'  column  in  the  1st  operation,)  are  24  tens,  and  the  1 

ten,  which  was  obtained  from  the  units,  are  25  tens,  equal  to 

5  tens,  which  write  in  the  tens'  place,  and  2  hundreds,  which 

write  in  the  hundreds'  place  ;  and  the  amount,  256  quarts, 

is  obtained  as  before. 

95.    Definitions  of  Terms,  and  the  Sign  for  Multipli- 
cation. 

A  product  is  a  number  produced  by  multiplication. 

A  multiplicand  is  a  number  to  be  multiplied. 

A  multiplier  is  a  number  showing  how  many  times  a  mul- 
tiplicand is  taken  to  form  a  product. 

Thus,  the  32  in  the  above  example,  is  the  multiplicand,  8 
the  multiplier,  and  256  is  the  product. 

The  multiplicand  and  multiplier  are  also  called  producers, 
ot  factors  of  their  product. 

Thus,  32  and  8  zx^  factors  of  256. 

X    The  oblique  cross  is  the   sign  for  multiplication.     Ii 


32 

32 

32 

8 

32 

32 

256 

32 

32 

32 

32 

MULTIPLICATION. 


25 


implies  mat  the  number  which  precedes  the  sign,  is  to  be 
multiplied  by  the  number  which  follows  it.  Thus,  32  X  S 
==  256,  which  is  read,  32  multiplied  by  8  equals  256,  or  8 
times  32  equals  256. 

26.   Multiplication  Table. 

In  order  to  perform  multiplication  with  facility,  you  will, 
before  attempting  further  progress,  correctly  ascertain,  and 
thoroughly  commit  to  memory,  the  product  of  each  combina- 
tion of  two  factors  in  the  following  table. 


2  times  2  equal 

3  X  2,  or  2  X  3  = 

4  X  2,  or  2  X  4  = 
5X2,  or  2X5  = 

6  X  2,  or  2  X  6  = 

7  X  2,  or  2  X  7  = 
8X2,  or2xS  = 
9  X  2,  or  2  X  9  = 
3  times  3  equal 
4X3,  or  3X4  = 

5  X  3,  or  3  X  5  = 

6  X  3,  or  3  X  6  = 


7X3,  or  3X7  = 

Sx  3,  or3  X  8  = 
9  X  3,  or  3  X  9  = 

4  times  4  equal 
5X  4,  or4x  5  = 

6  X  4,  or  4  X  6  = 

7  X  4,  or  4  X  7  = 
8X4,  or  4X8  = 
9  X  4,  or  4  X  9  = 

5  times  5  equal 

6  X  5,  or  5  X  6  = 

7  X  5,  or  5  X  7  = 


8  X  5,  or  5  X  8  = 

9  X  5,  or  5  X  9  = 

6  times  6  equal 

7  X  6,  or  6  X  7  = 

8  X  6,  or  6  X  8  ^ 

9  X  6,  or  6  X  9  — 

7  times  7  equal 
8X7,  or  7X8  = 
9X7,  or  7x9  = 

8  times  8  equal 

9  X  8,  or  8  X  9  = 
9  times  9  equal 


How  many  gallons 


87.   Model  of  a  Recitation. 

1.    In  one  hogshead  are  63  gallons, 
in  9  hogsheads? 

Since  there  are  63  gallons  in  1  hogshead,  in  9  hogsheads 

there  are  9  times  63  gallons,  which  is  obtained 

63  gallons,     by  multiplying  63  by  9.     Thus,  9  times  3 

9  units  are  27  units,  equal  to  7  units,  which 

write,  and  2  tens,  which  add  with  the  tens  ;   9 

567  gallons,     times  6  tens,  and  the  2  tens  obtained  from  the 

units,  are  5Q  tens,  equal  to  6  tens,  which  write, 

and  5  hundreds,  which  write  also,  giving  567  gallons  for  the 


28.   Exercises   in   Multiplying   when   the 
consists  of  but  one  figure. 


Multiplier 


In  like  manner,  solve  arid  explain  the  following  problems . 

1.  How  many  gallons  in  3  hogsheads  ? 

2.  How  many  gallons  in  7  hogsheads  ? 

3.  In  1  hour  are  60  minutes.  How  many  minutes  in  5  hours  ? 

3 


26 


ARITHMETIC. 


4.  How  many  minutes  in  8  hours  ? 

5.  If  100  cents  equal  1  dollar,  how  many  cents  are  equal 
to  6  dollars  ? 

6.  How  many  cents  in  4  dollars  ? 

7.  How  many  cents  in  9  dollars  ? 

8.  In  1  mile  are  320  rods.     How  many  rods  in  7  miles? 

9.  How  many  rods  in  5  miles  ? 

10.  If  2240  pounds  of  cotton  load  1  car,  how  many  pounds 
will  load  a  train  of  8  cars  ? 

11.  How  many  pounds  will  load  7  cars  ? 

12.  If  it  require  40000  inhabitants  to  send  1  representative 
to  Congress,  how  many  inhabitants  in  a  state  which  sends 
nine  representatives  ? 

13.  If  30000  persons  in  a  year  die  of  drunkenness,  how 
many  will  die  drunkards  in  the  next  5  years,  unless  people 
become  more  temperate  ? 

14.  If  each  of  these  drunkards  makes  seven  persons 
unhappy,  how  many  will  thus  be  made  unhappy  in  the  next 
5  years  by  their  drunkenness  ? 

15.  If  sound  move  1142  feet  in  a  second,  how  far  off  is 
the  thunder,  when  6  seconds  elapse  between  seeing  the  light- 
ning and  hearing  the  thunder  ? 

16.  If  the  salary  of  the  president  be  25000  dollars  a  year, 
how  much  has  been  paid  to  each  of  the  presidents  ? 

17.  What  is  the  product  of  2796  multiplied  by  4  ? 

18.  Multiply  675  by  5. 

19.  How  many  are  1789  X  7  ? 

39*    Composite  and  Prime  Numbers. 

1.  How  many  trees  in  an  orchard,  which  has  15  rows  of 
18  trees  each? 

Since  there  are  18  trees  in  1  row,  in  15  rows  there  will  be 
15  times  18  trees  ;  which  is  obtained  by  multiplying  18  by  15. 
This  multiplier,  consisting  of  two  figures,  presents  a  difficulty 
which,  however,  you  can  obviate  by  obtaining  a  part  of  the 
product  at  a  time ;  thus :  —  Since  in  15  rows  there  are  3 

times  5   rows,  multiply 

18  trees  in  1  row.  18  by  5  for  the  trees  in 

^  5  rows,  and  this  product 

90  trees  in  5  rows.  by  3  for  the  trees  in  3 

3  times    5,   or    15    rows, 

which  will  give  the  an* 

270  trees  in  3  times  5,  or  15  rows,     g^y^j.  required. 


MULTIPLICATION.  27 

A  number  which  is  composed  of  two,  or  more  factors,  as 
15  =  5  X  3,  or  42  ==  7  X  3  X  2,  &c.,  is  called  a  composite 
number. 

A  prime  number  is  a  number  which  has  no  factors,  except 
itself  and  unity;  as  1,  3,  5,  7,  11,  13,  17,  19,  23,  29,  &c. 

Hence  observe,  that,  when  the  multiplier  is  a  composite 
number,  the  product  may  be  obtained,  as  in  the  last  example, 
by  separating  the  multiplier  into  two,  or  more  factors,  and 
multiplying  first  by  one  factor,  then  that  product  by  aiwther 
factor,  and  so  on,  until  all  the  factors  have  been  used.  The 
last  product  will  be  the  product  required, 

30.  Model  of  a  Recitation. 

If  1  gallon  of  molasses  costs  42  cents,  what  will  be  the 
cost  of  1  hogshead  at  the  same  rate  ? 

Since  1  gallon  costs  42  cents,  1  hogshead,  which  is  63 
gallons,  will  cost  63  times  42  cents.  This  is  obtained  by 
separating  the  multiplier  into  its  factors,  9  X  7  =  63,  and 

42  cents,  the  cost  of  1  gallon.  i«"lt¥ying  first  by  9,  to 

Q                                  ®  obtain  the  cost  of  9  gai- 

Ions,  and  this  product  by 

378  cents,  the  cost  of  9  gallons.  7,  to  obtain  the  cost  of  7 

7  times  9  gallons,  or  63  gal- 

ocTc       *    *u         4.  rao     ii  lons;  which  is  2646  cents, 

2646  cents,  the  cost  of  63  ffallons.     ,u  ^     -    a 

'  ^  the  answer  required. 

31.  Exercises  in  Multiplying  by  Composite  Numbers. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  How  many  gallons  in  35  hogsheads  ? 

2.  How  many  gallons  in  45  hogsheads  ? 

3.  How  many  rods  in  21  miles,  there  being  320  in  one 
mile  ? 

4.  How  many  minutes  in  18  hours  ? 

5.  How  many  cents  in  42  dollars  ? 

6.  How  many  rods  in  63  miles  ? 

7.  What  would  28  bales  of  cotton  come  to,  at  75  dollars  a 
bale? 

8.  What  would  16  chests  of  tea  cost,  at  87  dollars  a  chest  ? 

9.  What  would  be  the  cost  of  a  drove  of  5Q  horses,  at  84 
dollars  a  piece  ? 

33*   Model  of  a  Recitation. 

How  many  are  24  times  27  ? 


5.  Multiply  4004  by  64. 

6.  Multiply  50000  by  35. 

7.  Multiply  908070  by  45. 

8.  Multiply  18273645  by  36. 


2S  ARITHMETIC. 

Since  6  X  4  =  24,  first  obtain  6  times  the  multiplicand, 

27  which  is  162,  then 

Q  4  times    this  pro- 

duct,  making  648, 

162  =  6  times  27.  which  is  4  times  6 

4  times,  or  24  times 

648  =  4  times  6  times,  or  24  times  27.     the  multiplicand,  as 

required. 

•I3.  Exercises  in  Multiplying  when  both  Factors  are 
Abstract  Numbers. 

In  like  manner^  solve  and  explain  the  following  problems, 

1.  What  is  81  times  47  ? 

2.  What  is  48  times  70  ? 

3.  How  much  is  79  X  54  ? ' 

4.  Multiply  123  by  72. 

34*   Model  of  a  Recitation. 

1.  What  would  10  cows  cost,  at  25  dollars  each  ? 

Since  1  cow  costs  25  dollars,  10  cows  cost  10  times  25 
dollars ;  the  amount  of  which  is  ascertained  by  annexing  a 
cipher  to  the  multiplicand,  making  250  dollars ;  for  now  the 
figures  of  the  multiplicand,  occupying  places  one  degree 
higher,  express  10  times  their  former  value,  (4:«) 

2.  If  128  dollars  be  paid  to  each  of  1000  men,  how  many 
dollars  would  they  all  receive  ? 

Since  1  man  would  receive  128  dollars,  1000  men  would 
receive  1000  times  128  dollars ;  the  amount  of  which  is 
ascertained  by  annexing  three  ciphers  to  the  multiplicand, 
making  128000  dollars ;  for  thus,  the  figures  of  the  multipli- 
cand are  made  to  occupy  places  three  degrees  higher,  and,  con- 
sequently, (4^  lOj)  express  1000  times  their  former  value. 

33,    Exercises    in    Multiplying    by    one    Unit    of   any 
Order. 

In  like  manner^  solve  and  explain  the  following  prohleTns. 

1.  What  will  10  yards  of  cloth  cost,  at  5  dollars  a  yard  ? 

2.  What  must  I  pay  for  100  sheep,  at  7  dollars  apiece  ? 

3.  What  would  be  the  cost  of  a  rail-road  100  miles  in 
length,  at  5796  dollars  a  mile  ? 

4.  What  would  be  the  price  of  10000  feet  of  boards  at  2 
cents  a  foot  ? 


MULTIPLICATION.  29 

11    What  is  the  stage  fare  for  1000  miles,  at  5  cents  a  mile  ? 


().  Multiply  161  by  10. 
V.  What  is  100  times  1728? 
8.  How  many  are  18  X  1000? 
i).  What  is  the  product  of  125 
and  1000  ? 


10.  Multiply  200  by  100. 

11.  Multiply  5000  by  100000. 

12.  Hdw   many   are    1020  X 

1000? 

13.  Multiply  1000  by  1000. 


«t6«   Model  of  a  Recitation. 

A  farmer  raised  84  bushels  of  potatoes  on  each  of  40  acres  ; 
'vhat  was  the  whole  number  of  bushels  ? 

Since  84  bushels  were  raised  on  1  acre,  on  40  acres  there 

were  40  times  84  bushels  raised.    The  multiplier  being  a  com- 

])osite  number,  (SOj)  whose  factors  are  4  and  10,  multiply 

o^  1     1    1  1  first  by  4,  to  ascertain  the  bushels 

84  bushels  on  1  acre.  a         ^i.      ^  u-  i  r  *u«*  ,.^^ 

j^pj  on  4  acres,  then  multiply  that  pro- 

duct  by.  10,  which  is  done  by  an- 

noam.    u  i         /in  nexin^  a  cipher,  (4.)  to  ascertain 

,3360  bushels  on  40  acres,     ^i     i,    i,  i        •    j  '    in  *•         a 

the  bushels  raised  on  10  times  4, 

or  40  acres,  which  gives  3360  bushels,  as  required. 

37*  Exercises  in  Multiplying  by  any  Number  of  Units 
OF  THE  same  Order. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  What  will  30  barrels  of  flour  come  to,  at  7  dollars  a 
barrel ? 

2.  If  it  take  20  men  6  days  to  do  a  job,  how  long  would 
it  take  1  man  to  do  it  ? 

3.  If  320  rods  make  a  mile,  how  many  rods  in  500  miles  ? 

4.  How  long  would  it  'take  1  man  to  do  what  40  men 
could  do  in  2  days  ? 

5.  How  much  is  300  times  125? 

6.  What  is  the  product  of  72 

and  900  ? 


7:  Multiply  1836  by  6000. 

8.  How  much  is  700x700? 

9.  Multiply  2500  by  2500. 


I 


518.   Model  of  a  Recitation. 

1.    In  1  quart  are  2  pints.     How  many  pints  in  67  quarts  ? 

Since  there  are  2  pints  in  1  quart,  in  67  quarts  there  will 
be  67  times  2  pints  ;  the  amount  of  which  maybe  ascertained 
by  multiplying  2  pints  by  67.  But  thd  multiplier,  67,  being 
a  prime  number,  (SO,)  presents  a  difficulty.  This,  however, 
you  can  obviate  by  taking  a  different  view  of  the  question. 
Thus,  since  there  are  2  pints  in  1  quart,  there  will  be  2  times 
3# 


30  ARITHMETIC. 

67  as  Tuany  pints  as  quarts^  the  amount  of  which 

2  may    be    ascertained    by    muhiplying    67    by 

2,  making    134    pints,   which    is   the    answer 

134  pints,  required. 

39*  Exercises  in  Changing  the  Order  of  the  Factors 

FOR  Multiplication. 

In  like  manner^  solve  and  explain  the  following  problems, 

1.  How  many  pints  in  29  quarts? 

2.  What  is  the  price  of  a  bushel  of  nuts,  at  6  cents  a  quart  ? 

3.  What  must  I  give  for  15  lemons,  at  4  cents  apiece  ? 

4.  If  a  man  plant  6  grains  of  corn  in  a  hill,  how  many 
grains  will  it  take  to  plant  a  field  having  75  rows  of  100  hills 
each? 

5.  What  would  an  ox,  weighing  873  pounds,  come  to,  at 
10  cents  a  pound  ? 

6.  If  27  men  receive  100  dollars  apiece,  how  much  do 
they  all  receive  ? 

7.  If  100  cents  make  a  dollar,  how  many  cents  in  47  dollars  ? 

8.  If  1000  mills  make  a  dollar,  how  many  mills  in  71 
dollars  ? 

9.  How  many  cents  in  53  dollars  ? 

10.  How  many  mills  in  53  dollars  ? 

11.  If  a  man  earn  10  dollars  a  week,  how  much  would 
he  earn  in  a  year,  which  is  52  weeks  ? 

12.  If  2  men  thresh  20  bushels  of  rye  in  a  day,  how  much 
would  they  thresh  in  23  days  ? 

13.  How  many  soldiers  in  a  brigade,  which  consists  of  32 
companies  of  60  soldiers  each  ? 

14.  How  many  pounds  of  beef  in  13  barrels  of  200  pounds 
each  ? 

15.  How  many  squares  on  a  chequer-board,  there  being  8 
rows  of  squares,  and  8  squares  in  each  row  ? 

16.  If  you  draw  straight  lines  across  your  slate,  both 
ways,  so  as  to  make  8  rows  of  squares  one  way,  and  12  rows 
the  other  way,  how  many  squares  would  there  be  ? 

1 7.  How  many  trees  in  an  orchard  which  has  41  rows  of 
40  trees  each  ? 

40.   Proof  of  Multiplication. 

You  may,  perhaps,  infer,  (265  38^)  that  the  product  of 
two  fajctors  is  the  same,  lohichsoever  he  made  the  vinltiplier* 


MULTIPLICATION.  31 

This  is  true ;  and  to  make  it  still  more  evident,  you  will  care- 
fully attend  to  the   following  demonstrations. 

3=1+1  +  1 

1^ 7 

7  times  3  =  21  =  7  +  7  +  7  =  3  times  7. 

Explanation :  7  times  3  is  the  same  as  7  times  each  unit 
In  3.  The  units  in  3  are  1  +  1  +  1,  which  nmltiplied  by  7 
give  7  +  7  +  7  =:  3  times  7. 

Generally  —  the  product  of  two  factors  is  ds  many  times 
the  multiplicand  as  there  are  units  in  the  multiplier,  (S4,) 
and  in  multiplying  we  multiply  each  unit  in  the  mvlti- 
plicand.  But  multiplying  one  unit,  gives  the  multiplier. 
Consequently,  multiplying^  each  unit  in  the  multiplicand, 
icill  give  as  many  times  the  multiplier  as  there  are  units  in 
the  multiplicand.  ^ 

Hence,  to  prove  the  correctness  of  an  operation  in  multipli- 
cation, make  the  multiplicand  the  multiplier,  and  repeat  the 
operation.     If  the  results  agree,  probably  both  are  correct. 

41.  Exercises  in  Proving  Multiplication. 

1.  Prove  that  5  times  3  is  equal  to  3  times  5. 

2.  Prove  that  6  times  5  must  be  equal  to  5  times  6. 

3.  Demonstrate  the  equality  of  6  X  3  and  3x6. 

4.  How  many  hills  in  a  potato-field  having  20  rows 
lengthwise,  and  16  rows  breadthwise  ? 

5.  How  many  hills  in  a  cornfield  having  50  hills  one 
Way,  and  25  hills  the  other  way? 

6.  Demonstrate  that  the  product  of  any  two  factors  will 
not  be  changed  by  changing  the  order  of  the  factors. 

42.  General  Explanation  of  Multiplication. 

1.  Mr.  Farmer  gave  67  dollars  an  acre  for  a  farm  of  222 
acres.     What  did  his  farm  cost  ? 

Since  1  acre  cost  67  dollars,  222  acres  must  have  cost  222 
times  67  dollars.  Neither  of  these  numbers  can  be  separated 
into  convenient  factors.  But  observe  that  the  multiplier, 
222  =  200  +  20  +  2.  Hence,  you-  may  multiply  by  these 
parts  of  the  multiplier,  separately,  and  then  add  the  three 
products.  This  will  give  as  many  times  the  multiplicand  as 
there  are  units  in  the  multiplier,  (^Ml,)  and  consequently,  the 
right  answer  to  the  question.      Thus  : 


32  AnnifiviKTir. 

67   dolh.  cost  of  1  acre.  (57   dolls,  cost  oi  1  iicre. 

200  20 


13400  dolls.  co8t  of  200  acres.     1340  dolls,  cost  of  20  acres. 


07  dolls,  cost  of  1  acre. 
134  dolls,  cost  of  2  acres. 


222 


134 
134 


131  dolls,  cost  of  2  txcvva. 

1310  dolls,  cost  of  20  acres. 

13400  dolls,  cost  of  200  acres. 

14874  dolls,  cost  of  222  acres,  the  answer  rctjuired. 

This  operation  may  Ix?  very  much  abridged.     Thus  : 
^.~     Having  written  the  w/iolr  nmltiplier  under  thf;  mulii 
plicand,  multi|>ly  lirst  hy  the  2  units,  ihcrj  l)y  iIh^  2 

tens,  or  20,  nnd  then  by  the  2  hundrrds,  or  200,  ar- 

^nM  ranging  the  j>rodurts  to^'etlier,  so  that  the  units  of  thv. 
same  orders  may  stand  in  the  same  roh/nins.  Multi- 
ply hy  the  2  uiiitH,  as  usual.     The  factors  of  20  beintf 

2  and  10,  nudtiply  hy  the  2,  and  make  this  product  10 

1dA7d     ^""^''**  ^^  larj^e,  (99,)  l>y  vvrilint:  it  one  degree  to  the 
^^^^     left,  (4.)     The  factors  of  200  heing  2  and  100,  multi- 
ply by  2,  and   make   this   product   100   timrs  as    large,  by 
writing  it  two  degrees  to  the  left,  (SM.)     1'l>'"  ^'nn  of  \\wne 
products  will  be  the  answer  recjuired. 
2.  Multiply  20003  by  1007. 
'>00(n     ^^^^^^  ^^^'^^'  '^'^*  multiplicand  7  times,  then  1  thou- 
^1007     ^'^'^^^    times.     To    nmltiply   by   the    1000,  write 

onc(^  the  multiplicund,  three  degrees  to  the  left, 

140021      (JM.)      The  three  ciphers  neetj  not  Ik?  armexed  ; 
20003  for,  without  them,  each  figun^  of  this  product  will 

om^QH^T     ^^^'  ^^^  ^^*^*  column  of  units  of  its  own  order,  and 
-iU14.lU^l      ^i^j^rofore  will  be  added  in  the  right  plare. 

4S«     MoDKL    OK    A    RkCITATION. 

How  nmch  is  30r)08  times  403070  ? 

/in*^n7n     l^^'r(^  the  nndtiplicand  is  to  be  taken  S  times, 

on^nfi     500  times,  and  30000  times.     Multiply  by  the 
.5U0UO     Q      rp^^^^  p^^,^^^^^  ^^^^3^  l^^,j^^  ^  ^jj^,  j^^^  ^^jj^ 


3224/500  ^^P'y  ^^y  ^'  '^^''^  make  this  j)ro(luct  100  limes 

201/)3f50  ^^  I'vrge,  by  writing  it  iip  two  degrees.     The 

1209210  factors  of  30000  })eing  3  and  lOOOO,  multiply 

by  3,  and  make  this  prod  tic  t  10000  times  as 

I2296a59660  large,  hy  writing  it  up  four  degrees.     Th« 


MULTIPLICATION.  U 

Bum  of  these  partial  products,  thus  arranged,  will  be  Uie  pro- 
duct lequired. 

^-l.     OnSKIlVATION. 

Obskrve,  that  in  these  operations  (49,  43)  the  midtipli' 
Zand  is  multiplied  by  each  digit  in  the  multiplier^  that  the 
first  figure  in  each  partial  product  is  of  the  same  denomination 
as  the  mulliplyiiig  figure y  and  that  the  sum  of  Oie  partial  pro- 
ducts is  the  product  required. 

4A.   Exercises  in  Multiplication. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  A  man  travelled  26  days,  at  the  rate  of  47  miles  a  day. 
How  fnr  did  he  travel  ? 

2.  If  a  chaise  wheel  turn  round  346  times  in  1  mile,  how 
many  times  will  it  revolve  in  the  25  miles  from  Boston  to 
Lowell  ( 

3.  How  much  money  would  be  required  to  pay  37  men 
7/5  (lolhirs  apiece  ? 

4.  Wlijit  must  I  pay  for  29  fat  oxen,  at  43  dollars  apiece  ? 

5.  What  will  97  tons  of  iron  come  to,  at  57  dollars  a  ton  ? 

6.  If  a  vessel  sail  1«58  miles  a  day,  how  far  would  it  sail 
ill  [hv.  month  of  April  ? 

7.  If  786  yards  of^cloth  are  made,  daily,  in  a  factory  which 
luris  313  days  a  year,  what  is  made  annually  in  that  fac- 
tory? , 

a.  How  much  wheat  can  be  raised  on  95  acres,  at  38 
bushels  an  acre  ? 

9.  Multiply  1728  by  144. 

10.  How  much  is  4004  times  999  ? 

11.  What  is  the  product  of  6075  and  67  ? 

12.  How  much  is  160012  X  333? 

13.  Multiply  1H36  hy  1010. 

14.  Mnliiply  1111  by  2222. 

15.  Multiply  2222  by  1111. 

16.  Multiply  3000024  by  309. 

17.  Multi])ly  309  by  3000024. 

40*   Model  of  a  Recitation. 

What  is  the  product  of  32000  and  2300  ? 


34  ARITHMETIC. 

32000  '^^^  factors  of  2300  being  23  and  100,  multiply 

2S00  ^y  ^^^  placing  it  under  the  digits  of  the  multiph- 

cand,  and  multiplying  without  regard  to  the  ci- 

q^  phers  on  the  right.  This  gives  736.  But,  since 
^A  23  times  32  units  of  any  order  will  be  736  units 
of  the  sa7?i€  order,  as  surely  as  23  times  32  things 

7*^600000  ^^  ^^y  ^^^^^  ^^'^^^  §^^'^  ^^^  things  of  the  same 
kind ;  23  times  32  thousands  will  be  736  thou- 
sands. Therefore  annex  three  ciphers,  (845)  that  it  may 
have  the  thousands'  place  ;  then  annex  two  ciphers  more, 
(30^)  to  multiply  by  the  other  factor  in  the  multiplier ; 
which  gives  the  product  required. 

47.  Observation. 

Observe,  that,  by  this  process,  (4O5)  as  many  ciphers  wiU 
be  annexed  to  the  product  of  the  digits  as  there  are  on  the 
right  of  both  factors. 

48.  Exercises  in  Multiplying,  when  the  Factors  ex- 

press Units  of  the  higher  Orders. 

In  like  manner y' solve  and  explain  the  following  problems. 

1.  How  far  is  it  from  Boston  to  Liverpool,  if  a  vessel  sail 
from  Boston  at  the  rate  of  150  miles  a  day,  and  arrive  at 
Liverpool  in  20  days  ? 

2.  How  far  from  the  earth  to  the  sun,  if  it  take  light  480 
^seconds  to  come  from  the  sun,  at  200000  miles  a  second  ? 

3.  What  is  the  capital  of  Boston  Bank,  there  being  12000 
shares,  at  50  dollars  a  share  ? 

4.  What  is  the  capital  of  Massachusetts  Bank,  there  being 
8200  shares,  at  250  dollars  a  share  ? 

5.  State  Bank  has  30000  shares,  at  60  dollars  each.  What 
is  its  capital  ? 

6.  If  Massachusetts'  house  of  representatives  has  500  mem- 
bers, and  a  session  lasts  90  days  ;  how  much  money  would 
it  take  to  pay  2  dollars  a  day  to  each  member  ? 

7.  How  many  weekly  newspapers  will  it  require  to  furnish 
30000  subscribers  one  year  ? 

S.  What  would  be  the  cost  of  a  railroad,  40  miles  in 
length,  at  40000  dollars  a  mile  ? 

9.  Multiply  740  by  6050. 

10.  Multiply  6050  by  740. 


MULTIPLICATION.  3S 

low  many  are  3400  times  390  ? 

12.  How  many  are  390  times  3400  ? 

13.  What  is  the  product  of  140  multiplied  by  140  ? 

14.  Multiply  1600  by  itself. 

15.  55500  X  4400  is  how  much  ? 

16.  1910  X  170  are  how  many  ? 

17.  If  the  multiplicand  be  160000,  and  the  multiplier  2400, 
what  will  be  the  product  ? 

18.  121212x8080  =  ? 

49.   General  Exercises  in  Addition  and  Multiplication. 

1.  How  many  months  was  Andrew  Jackson  president? 

2.  How  many  months  was  John  Quincy  Adams  president  ? 

3.  How  many  pounds  of  pork  on  150  wagons,  each  loaded 
with  6  barrels,  with  200  pounds  in  a  barrel  ? 

4.  If  a  house  have  20  windows,  of  24  panes  each,  how 
many  panes  in  all  the  windows  ? 

5.  W  hat  number  is  9000  times  165  ? 

6.  What  number  contains  144  twelve  times  ? 

'  7.  What  number  contains  one  thousand  and  fifteen  607 
times  ? 

8.  What  would  be  the  sum  of  457  set  down  ten  thousand 
times,  and  added  up  ? 

9.  What  is  the  cost  of  a  road  40  miles  long,  of  which  one 
half  cost  1750  dollars  a  mile,  and  the  other  half,  1800  dollars 
a  mile  ? 

10.  If  a  quantity  of  provisions  would  last  500  men  30 
days,  how  long  would  it  last  1  man  ? 

11.  How  many  men  would  consume  in  1  day  what  would 
last  500  men  30  days  ? 

12.  If  a  bushel  of  wheat  afford  70  ten-cent  loaves,  how 
many  cent  loaves  may  be  obtained  from  it  ? 

13.  How  many  yards  of  cloth,  1  quarter  wide,  are  equal 
to  27  yards  5  quarters  wide  ? 

14.  How  long  would  it  take  a  man,  working  1  hour  a  day, 
to  do  what  he  could  in  26  days,  working  12  hours  a  day  ? 

15.  If  a  boy  attend  school  constantly  3  terms  of  12  weeks, 
and  1  term  of  1 1  weeks  ;  how  many  hours  is  he  in  school,  at 
33  hours  a  week  ? 

16.  How  many  strokes  will  the  city  clock  strike  in  the 
month  of  June  ? 

17.  If  it  take  594  bricks  to  pave  I  rod  of  side-walk,  how 
many  would  it  take  to  pave  a  walk  a  mile  long  ? 


36  ARITHMETIC. 

18.  What  are  a  man's  annual  expenses,  who  pays  3  dollars 
a  week  for  board,  6  dollars  a  month  for  clothes,  10  dollars  a 
quarter  for  travelling  expenses,  1  dollar  a  week  for  benevolent 
purposes,  and  for  other  items  75  dollars  ? 

19.  What  is  a  man's  income,  who  receives  a  salary  of  15 
dollars  a  week,  and  10  dollars  a  month  interest  money  ? 

20.  What  is  the  value  of  a  drove  of  cattle,  consisting  of 
12  oxen  at  55  dollars  apiece,  15  cows  at  30  dollars  apiece, 
18  heifers  at  16  dollars  apiece,  and  14  yearlings  at  10  dollars 
apiece  ?  , 

21.  What  is  the  amount  of  the  following  bill  ? 

Boston,  April  25,  1846. 
Mr.  John  Merchant, 

Bought  of  Charles  Wholesale, 
27  yards  of  Black  Broadcloth,  at  $6  a  yard, 
25     "  Blue  "  u      7      u 

18     **  Drab  Cassimere,     "      3      " 

24  Vest  Patterns,  *'      2  a  pattern. 


Received  payment, 

Charles  Wholesale. 

22.  What  is  the  foot  of  the  following  bill  ? 

Hanks,  Harris  &  Co.  »^^^^^'  ^P^^^  ^^^  ^^^- 

Bought  of  Burt  &  Townsend, 
1200  pairs  Boys'  Shoes,  <®  $  1  per  pair, 

400     "     Men's     "       ©    2       "  .       . 

600    *  "      Boots,  (©    3       "  .       .* 


23.  What  is  the  foot  of  the  following  accbunt  ? 

Mr.  Isaac  Speculator, 

1846.  To  Jonathan  Farmer,  Dr. 

Jan.    31.    To  17  Cords  Wood,     (8)  $7  per  cord, 
Aug.    1.     "     9  Tons  Hay,         (b  15   "    ton, 
Oct.    12.     •'    10  Loads  Potatoes,  (S>    8  ''    load, 

"      15.     '♦    18  Barrels  Apples,  <®    2   ♦'    barrel. 


24.  How  many  scholars  can  a  school-room  accommodate, 
in  which  are  4  divisions  of  seats,  11  rows  of  seats  in  each 
division,  and  6  seats  in  a  row  ? 


MULTIPLICATION.  37 

25.  How  many  are  4  X  H  X  6  ? 

26.  How  many  seats  in  a  church,  in  which  the  body  pews 
are  in  4  roVs  of  18  pews  each,  the  wall  pews  in  2  rows  of 
24  pews  each,  and  the  gallery  pews  in  12  rows  of  4  pews 
each,  there  being  6  seats  in  each  pew  ? 

27.  How  many  shingles  will  cover  the  roof  of  a  house, 
each  of  the  two  sides  being  32  feet  long  and  16  feet  wide  ; 
if  it  take  3  shingles  to  extend  a  foot  in  each  direction  ? 

28.  What  is  the  product  of  32  X  3  X  16  X  3  X  2  ? 

29.  If  the  earth  move  in  its  orbit  68000  miles  an  hour, 
how  far  does  it  move  in  24  hours  ? 

30.  How  far  in  its  orbit  does  the  earth  move  in  the  month 
of  February  ? 

31.  How  far  does  the  earth  move  in  the  4  months  which 
have  30  days  each  ? 

32.  How  far  does  the  earth  move  in  the  7  months  which 
have  31  days  each  ? 

33.  How  many  miles  does  the  earth  move  in  a  year,  as 
shown  in  the  last  three  problems  ? 

34.  If  the  moon  is  240000  miles  distant,  and  the  sun  is 
400  times  as  far  ofl',  what  is  the  distance  of  the  sun  ? 

35.  What  number  is  that  whose  factors  are  3,  5,  7  ? 

36.  What  is  the  product  of  the  first  ten  prime  numbers  ? 

37.  What  sum  of  money  must  be  divided  among  27  men, 
so  that  each  man  may  receive  115  dollars  ? 

38.  Two  men  depart,  in  opposite  directions,  from  the  same 
place,  one  at  the  rate  of  27,  and  the  other  31  miles  a  day. 
How  far  are  they  apart  in  a  week  ? 

39.  Two  men  depart,  in  the  same  direction,  from  the  same 
place  ;  but  one  travels  10  miles  a  day  farther  than  the  other. 
How  far  apart  are  they  in  a  week  ? 

40.  The  product  of  two  equal  factors  being  called  the  secona 
potaer,  or  square  of  that  repeated  factor,  what  is  the  second 
power  of  12  ?  Ans.  12  X  12  =  144. 

41.  What  is  the  second  power  of  15  ? 

42.  What  is  the  second  power  of  30  ? 

43.  What  IS  the  square  of  50  ? 

44.  What  is  the  square  of  100  ? 

45.  The  product  of  three  equal  factors  being  called  the 
third  power,  or  cube  of  that  repeated  factor,  what  is  the  cube 
of  12 «  Ans.  12  X  12  X  12  =.  1728. 

4 


38 


ARITHMETIC. 


46.  What  is  the  cube  of  lo  ? 

47.  What  is  the  third  power  of  9  ? 

48.  What  is  third  power  of  25  ?  • 

49.  The  product  of  four  equal  factors  being  called  the 
fourth  power  of  that  repeated  factor,  what  is  the  fourth 
power  of  3  ?     Answer,  3x3x3x3  =  81. 

50.  What  is  the  fourth  power  of  5  ? 

51.  Any  number  being  the  first  power  of  itself,  what  are 
the  first  ten  powers  of  2  ? 

52.  What  are  the  first  ten  powers  of  10  ? 

53.  Multiply  144  by  the  third  power  of  10. 

54.  Multiply  18  by  the  fifth  power  of  10. 
^^,   Multiply  500  by  the  second  power  of  10. 


IV.   SUBTRACTION. 

SO.   The  Principles  of  Subtraction  Illustrated. 

In  Numeration  (2)  you  were  taught  that  the  addition  of 
one  unit  to  any  number,  formed  the  next  larger  number. 

Hence,  it  follows  that  taking  one  unit  from  any  htimber, 
leaves  the  next  smaller  number. 

In  Addition  you  were  taught  that  two  or  more  numbers, 
consisting  of  any  number  of  units,  could  be  united  into  one 
larger  number,  equal  to  their  sum. 

Hence,  it  follows  that  any  number  can  he  separated  into 
two  or  more  smaller  numbers^  the  sum  of  lohich  equals  the 
original  number. 

The  father  of  John  and  Henr^^  promised  to  give  them  10 
cents ;  but,  as  John  was  the  older  boy,  he  should  have  7 
cents,  and  Henry  might  have  the  remainder  of  them. 

Henry,  in  trying  to  make  his  part  as  many  as  possible, 
studied  out  these  curious  questions. 

1.  Hoio  many  ivill  remain,  when  John  has  taken  7  from 
the  10  cents? 

2.  How  many  more  are  the  whole  10,  than  John's  7  cents  ? 

3.  How  many  less  than  the  whole  10,  are  John's  7  cents  ? 

4.  Hoio  many  must  he  added  to  John's  7,  to  make  the 
whole  10  cents  ? 

6.  How  many  must  I  take  from  the  whole  10,  to  leave 
John's  7  cents  ? 


SUBTRACTION.  39 


6.  What  is  the  difference  between  John's  part,  and  the 
whole  10  cents  ? 

7.  What  is  tke  difference  between  the  whole  10  cents,  and 
John's  part  ? 

8.  If  10  cents  are  separated  mto  two  parts,  one  of  which  is 
7,  what  is  the  other  part  1 

But  he  found  that,  to  answer  all  his  questions,  he  had 
only  to  take  7  from  10,  and  that,  in  every  case,  only  3  cents 
remained  for  his  part. 

til.     Definitions    of    Terms,   and    the    Sign    for    Sub- 
traction. 

Subtraction  is  the  taking  from  a  number. 

Minuend  is  a  given  number  to  be  diminished  hy  subtraction. 

Subtrahend  is  a  given  number  to  be  subtracted. 

By  subtraction  the  minuend  is  separated  into  two  parts, 
one  of  which  equals  the  subtrahend. 

To  ascertain  the  other  part,  is  the  purpose  of  the  opera- 
tion. This  is  done  by  taking  the  subtrahend  from  the 
minuend.  The  number  which  is  left  is  the  part  required,  and 
is  called  the  Remainder.  It  is  the  Difference  between  the 
minuend  and  subtrahend. 

Observe,  in  the  questions  above,  (SOj)  that  10  is  the  given 
number  to  be  separated  into  two  parts,  and,  therefore,  is  the 
Minuend  ;  that  7  is  the  given  part  of  the  minuend,  and,  there- 
fore, is  the  Subtrahend  ;  that  3  is  the  other  part,  or  Remainder 
of  the  minuend,  and,  that  the  two  parts  of  the  minuend, 
7  -(-  3  =  10,  the  whole  minuend. 

Observe  also,  in  these  questions,  the  different  uses  of  sub- 
traction. 

—  One  horizontal  line  is  the  sign  for  subtraction.  It 
implies  that  the  number  which  follows  the  sign,  is  to  be  taken 
from  what  precedes  it,  thus :  10  —  7  =  3,  which  is  read,  10 
minus  7  equals  3.  Minus  is  the  Latin  word  for  less,  and, 
here,  means  the  same  as  if  you  should  say  10  less  7,  or  7  less 
than  10  equals  3.  10  is  the  minuend,  7  the  subtrahend ; 
and  3  is  the  difference. 

52.    Subtraction  Table. 

In  order  to  perform  subtraction  with  facility,  you  will, 
before  attempting  further   progress,  correctly  ascertain,  and 


40 


ARITHMETIC. 


thoroughly  commit  to  memory,  the  difference   between  the 
two  numbers  of  each  combination  in  the  following  table. 


2  —  2  = 

3  —  2  = 

4  —  2  = 

5  —  2  = 

6  —  2  = 

7  —  2  = 

8  —  2  = 

9  —  2  = 

10  —  2  = 

11  —  2  = 

6  —  6  = 

7  —  6  = 

8  —  6  = 

9  —  6  = 

10  —  6  = 

11  —  6  = 

12  —  6  = 

13  —  6  = 

14  —  6  = 

15  —  6  = 


3  —  3  = 

4  —  3  = 

5  —  3  = 

6  —  3  = 

7  —  3  = 

8  —  3  = 

9  —  3  = 

10  —  3  = 

11  —  3  = 

12  —  3  = 

7  —  7  = 

8  —  7  = 

9  —  7  = 

10  —  7  = 

11  —  7  = 

12  —  7  = 

13  —  7  = 

14  —  7  = 

15  —  7  = 

16  —  7  = 


4  —  4  = 

5  —  4  = 

6  —  4  = 

7  —  4  = 

8  —  4  = 

9  —  4  = 

10  —  4  = 

11  —  4  = 

12  —  4  = 

13  —  4  = 

8  —  8  = 

9  —  8  = 

10  —  8  = 

11  —  8  = 

12  —  8  = 

13  —  8  = 

14  —  8  = 

15  —  8  = 

16  —  8  = 

17  —  8  = 


6  —  !^z=z 

6  —  5  = 

7  —  5  = 

8  —  5  = 

9  —  5  = 

10  —  5  = 

11  —  5  = 

12  —  5  = 

13  —  5  = 

14  —  5  = 

9  —  9  = 

10  —  9  = 

11  —  9  = 

12  —  9  = 

13  —  9  = 

14  —  9  = 

15  —  9  = 

16  —  9  = 

17  —  9  = 

18  —  9  = 


53*   Model  of  a  Recitation. 

1.  A  man  bought  a  farm  for  2325  dollars,  and  sold  it  for 
2548  dollars.     How  many  dollars  did  he  gain  ? 

He  gained  the  difference  between  what  he  gave,  and  what 
he  received  for  his  farm. 

Here,  2548  is  the  minuend,  (as  the  larger  of  the  two  given 
numbers,  when  there  is  any  difference  between  them,  is 
always  the  minuend,)  and  2325  is  the  subtrahend.  It  will  be 
most  convenient  to  take  the  units  of  each  order  from  units  of 
the  same  order,  beginning  with  the 
lowest.  Therefore,  write  the  sub- 
trahend under  the  minuend,  placing 
the  units  of  each  order  under  those 
of  the  same  order.  Take  5  units 
from  8  units,  and  3  units  remain, 
which  write  in  the  units'  place ;  2  tens  from  4  tens,  2  tens 
remain,  which  write  in  the  tens'  place ;  3  hundreds  from  5 
hundreds,  2  hundreds  remain,  which  write  in  the  hundreds' 


2548  Minuend. 
2325  Subtrahend. 

223  Eemainder, 


SUBTRACTION.  41 

place ;  and  2  thousands  from  2  thousands,  nothing  reuains. 
Consequently,  223  dollars  is  the  answer  required. 

•I4«   Proof  of  Subtraction. 

To  prove  the  correctness  of  this,  or  any  operation  in  sub- 
traction, add  together  the  remainder  and  subtrahend.  If 
this  sum  agree  with  the  minuend,  probably  the  operation  is 
correct";  for  the  remainder  and  subtrahend,  being  the  two 
parts  into  which  the  minuend  is  separated,  the  reunion  of 
these  parts  ought  to  reproduce  the  minuend. 

ti^.  Exercises  in  Subtracting  when  no  Figure  of  the 
Subtrahend  exceeds  the  Corresponding  Figure  of 
THE  Minuend. 

In  like  Tnanner^  solve  and  explain  the  following  problems. 

1.  Charles  having  25  cents,  gave  12  of  them  for  a  book. 
How  many  cents  had  he  left? 

2.  Charles  paid  25  cents  for  a  book  and  slate,  13  cents 
was  the  price  of  the  slate,  what  was  the  price  of  the  book  ? 

3.  John  said  he  was  25  years  younger  than  his  father,  who 
was  37  years  old.     How  old  was  the  boy  ? 

4.  A  merchant  35  years  old,  had  traded  14  years.  How 
old  was  he  when  he  commenced  business  ? 

5.  In  a  school  of  84  scholars,  only  33  are  girls.  How 
many  boys  in  that  school  ? 

6.  A  man  sold  a  chaise  and  harness  for  198  dollars ;  but 
the  price  of  the  chaise  was  163  dollars.  What  was  the  price 
of  the  harness  ? 

7.  A  house  and  the  land  on  which  it  stood  cost  2350  dol- 
lars ;  but  the  house  cost  all  but  350  dollars.  What  was  the 
cost  of  the  house  ? 

8.  If  I  deposite  in  a  bank  1675  dollars,  and  afterv/ards 
draw  out  1000,  how  much  have  I  then  remaining  in  the 
bank? 

9.  Mr.  Walkers  farm  is  worth  3000  dollars,  and  Mr. 
Dole's  farm  is  worth  2000  dollars ;  if  they  exchange  farms, 
what  should  Mr.  Dole  pay  Mr.  Walker  ? 

10.  What  is  the  difference  between  5643  and  643  ? 

11.  How  much  more  is  12345  than  2040? 

12.  How  much  less  is  1620  than  1840  ? 
-^  13.  Subtract  203040  from  516273. 

1^    '* 


\ 


42  ARITHMETIC. 

tl6*   Model  of  a  Recitation. 

1.  A  man  paid  85  dollars  for  a  watch  ;  but  was  obliged  to 
sell  it  for  67  dollars.     What  was  his  loss  ? 

He  lost  the  difference  between  what  he  gave,  and  what  he 
received  for  his  watch. 

Arrange  the  numbers  and  proceed  as  before  directed. 
85  7  units,  however,  cannot  be  taken  from  5  units. 
67         But,  since  (10)   1  unit  of  any  order,  equals    10 

—  units  of  the  next  lower  order,  reduce  (208)  one  of 
18         the  8  tens  to  units,  making  10  units,  which,  united 

with  the  5  units,  make  15  units,  from  which  take 
the  7  units,  8  units  remain,  which  write  in  the  units'  place, 
and  take  the  6  iens^not  from  8  tens,  for  one  of  them  has  been 
reduced  to  units  and  disposed  of;  but  take  6  tens  from  7  tens, 
1  ten  remains,  which  write  in  the  tens'  place.  Hence,  18 
dollars  is  the  answer  required. 

2.  What  is  the  difference  between  9342  and  5739  ? 
Reduce  one  of  the    4  tens  to  units,  making   ten   units, 

which,  united  with  the  2  units,  make  12  units, 
9342  from  which  take  the  9  units,  3  units  remain ; 
5739         take  the  3  tens  of  the  subtrahend  from  the  other 

3  tens  of  the  minuend,  nothing  remains ;  there- 

3603         fore,  write  a  cipher  in  the  tens'  place ;   reduce 

one  of  the  9  thousands  to  hundreds,  making  10 
hundreds,  which,  united  with  the  3  hundreds,  make  13  hun- 
dreds, from  which  take  the  7  hundreds,  6  hundreds  remain ; 
take  the  5  thousands  from  the  other  8  thousands,  3  thou- 
sands remain.     Hence,  the  whole  difference  is  3603. 

57.  Exercises  in  Subtracting,  when  some  Figures  .of 
THE  Subtrahend  exceed  the  Corresponding  Figures 
OF  the  Minuend. 

In  like  manner,  solve  and  explain  the  folUrwing  problems. 

1.  A  man  gave  5  dollars  for  a  hat,  and  20  dollars  for  a 
coat.     How  much  less  did  his  hat  cost  than  his  coat  ? 

2.  Dr.  Franklin  died  A.  D.  1790,  and  was  84  years  old. 
In  what  year  was  he  born  ? 

3.  George  Washington  was  born  A.  D.  1732,  and  died  in 
1799.     How  old  was  he  when  he  died  ? 

4.  The  Puritans  landed  at  Plymouth  in  1620.   How  many 
years  since  ? 


SUBTRACTION,  43 

5.  How  long  since  Columbus  discovered  America  in  1492  ? 

6.  How  many  years  since  the  declaration  of  Independence 
by  the  United  States  in  1776  ? 

7.  The  Rocky  Mountains  are  12500,  and  the  Andes  21440 
feet  high ;  how  much  higher  are  the  Andes  than  the  Rocky 
Mountains  ? 

8.  The  Mississippi  river  is  3600  miles  long,  and  the 
Missouri  river  is  4500  miles  long  ;  how  much  longer  is  the 
latter  than  the  former  ? 

9.  In  Massachusetts  are  7800  square  miles,  and  in  New 
Hampshire  9491  ;  how  much  more  land  in  New  Hampshire 
than  in  Massachusetts  ? 

10.  How^much  larger  is  New  York,  which  contains 
46085  square  miles,  than  Massachusetts,  which  has  7800 
square  miles  ? 

11.  Subtract  147  from  222. 

12.  From  671  take  584. 

13.  How  much  is  746—475? 

14.  What  must  be  added  to  999,  to  make  1492  ? 

15.  What  must  be  subtracted  from  1840,  to  leave  1776  ? 

58.   Model  of  a  Recitation. 

^1.  A  man  obtained  at  a  bank,  300  dollars,  but  at  the  same 
time,  he  paid  back  18  dollars  for  interest ;  how  many  dollars 
had  he  left? 

He  had  left  the  difference  between  what  he  received  and 
what  he  paid  back,  which  is  ascertained  by  subtracting  18 
from  300. 

Here  there  are  no  units  from  which  to  take  the  8  units, 

neither  is  there  any  ten  to  reduce  to  units ;    there- 

^f\r\         fore,  reduce  one  of  the  3  hundreds  to  tens,  (SQ^) 

^^         making  10  tens ;  leaving  9  of  these  tens,  reduce 

the  other   to  units,  making  10  units,  from  which 

ooQ         take    the   8  units;  2  units  remain.     Take  the  1 

ten  in  the  subtrahend,  from  those  9  tens  that  you 

left  unused  ;  8  tens  remain.     There  is  nothing  to 

take  from  the  other  2  hundreds ;  therefore,  write  them  in  the 

hundreds'  place.     Hence,  282  dollars  is  the  answer  required. 

2.    Subtract  30206,  from  5000000. 


44 


ARITHMETIC. 


Reduce  one  of  the  5  millions  to  hundred-thousands,  making 

10  ;  one  of  which,  (leaving  9,)  reduce  to  ten- 

5000000         thousands,  making  10  ;  one  of  which,  (leaving 

80206         9,)  reduce  to  thousands,  making  10 ;  one  of 

which,  (leaving  9,)  reduce  to  hundreds,  making 


4969794  10;  one  of  which,  (leaving  9,)  reduce  to  tens, 
making  10  ;  one  of  which,  (leaving  9,)  reduce 
to  units,  making  10  units,  from  which  subtract  the  6  units ; 
4  units  remain.  Subtract  the  other  figures  of  tbe  subtrahend 
from  the  9s  that  were  left ;  saying,  cipher  from  9  .  tens 
leaves  9  tens ;  2  hundreds  from  9  hundreds  leaves  7  hun- 
dreds ;  cipher  from  9  thousands  leaves  9  thousands  ;  3  ten- 
thousands  from  9  ten-thousands  leaves  6  t(^-thousands  ; 
blank  from  9  hundred-thousands  leaves  9  hundred-thousands  ; 
and  blank  from  4  millions  leaves  4  millions.  Hence,  the 
whole  remainder  is  4969794. 

59.  Observation. 

Observe,  in  these  operations^  that  the  units  of  each  order 
in  the  subtrahend^  beginning  with  the  lowest^  are  subtracted 
from  the  units  of  the  same  order ^  in  the  minuend^  when 
possible ;  otherwise^  one  of  the  units  expressed  by  the  next 
higher  digit  iii  the  minuend,  is  mentally  reduced  (leaving 
95  in  the  intervening  places)  to  the  order  of  the  deficient 
figure,  and  united  with  it,  ivhen  the  subtraction  is  made 
from  what  then  remains  hi  the  several  places  of  the  min^ 
uend. 

60.  General  Exercises  in  Subtraction. 

I?i  like  manner,  solve  and  explain  the  following  problems, 

1.  The  top  of  a  flag-staff,  25  feet  long,  which  was  fasten- 
ed to  the  top  of  a  liberty-pole,  was  104  feet  high  ;  how  high 
was  the  liberty-pole  ? 

2.  If  17  feet  should  be  broken  from  the  top  of  a  tree,  100 
feet  high,  how  high  would  be  the  stump  ? 

3.  The  bell  on  a  church  is  75  feet  from  the  ground,  but 
the  vane  is  102  feet  from  the  ground ;  how  many  feet  from 
the  bell  to  the  vane  ^ 

4.  If  the  Creation  was  4004  years  B.  C,  and  the  Deluge 
234S  years  B.  C,  how  man}^  years  from  the  Creation  to  the 
Deluge  ? 

5.  How  many  years  from  the  Creation,  4004  years  B.  C. 
was  Saul  made  the  first  king  over  Israel,  in  1095,  B.  C.  ? 


SUBTRACTION.  45 

6.  In  1820,  New  Orleans  had  27176  inhabitants;  in 
!.S25,  35000  inhabitants;  what  was  the  increase  in  five 
•'^ears  ? 

7.  In  A.  D.  1825,  New  Orleans  had  35000  inhabitants ;  in 
.830,  46310;  what  was  the  increase  in  five  years  ? 

8.  In  A.  D.  1830,  New  Orleans  had  46310  inhabitants ; 
in  1835,  60000;  what  was  the  increase  in  these  five  years? 

9.  In  A.  D.  1835,  New  Orleans  had  60000  inhabitants ; 
Jind  Charleston,  S.  C.,  had  34500;  how  many  more  inhabi- 
tants in  New  Orleans,  than  in  Charleston,  S.  C,  in  1835? 

10.  In  A.D.  1820,  Philadelphia  had  119325  inhabitants; 
in  1825,  140000 ;  what  was  the  increase  in  these  five  years  ? 

11.  In  A.  D.  1825,  Philadelphia  had  140000  inhabitants; 
jn  1830,  167811 ;  what  was  the  increase  in  these  ^ve  years  ? 

12.  In  A.D.  1830,  Philadelphia  had  167811  inhabitants; 
:.n  1835,  200000  ;  what  was  the  increase  in  these  five  years  ? 

13.  In  A.  D.  1820,  Boston  had  43298  inhabitants ;  in 
1825,  58277  ;  what  was  the  increase  in  these  five  years  ? 

14.  In  A.D.  1825,  Boston  had  58277  inhabitants;  in 
:1830,  61381 ;  what  was  the  increase  in  these  five  years  ? 

15.  In  A.D.  1830,  Boston  had  61381  inhabitants;  in 
1835,  78613 ;  what  was  the  increase  in  these  five  years  ? 

16.  In  A.  D.  1820,  New  York  city  had  123706  inhabitants ; 
in  1830,  203007 ;  what  was  the  increase  in  these  ten  years  ? 

17.  In  A.D.  1835,  New  York  city  had  269873  inhabi- 
tants; Boston  had  78613;  how  many  more  inhabitants  had 
New  York  than  Boston  ? 

18.  How  much  farther  through  the  middle  of  the  sun  than 
through  the  middle  of  the  earth ;  the  former  being  883217 
miles,  and  the  latter  being  7916  miles? 

19.  What  is  the  diiTerence  between  the  diameters  of  the 
Earth  and  Jupiter ;  the  former  being  7916  miles,  and  the 
latter  89170  miles  ? 

20.  How  much  faster  does  the  Earth  move  than  Jupiter ; 
the  former  moving  68000  miles  an  hour,  the  latter  30000 
miles  an  hour  ? 

21.  How  much  is  1000  —  999  ? 

22.  How  much  more  is  380064  than  87065  ? 

23.  How  much  smaller  is  8756  than  37005078? 

24.  How  much  must  you  add  to  7643,  to  make  16487  ? 

25.  How  much  must  you  subtract  from  2483,  to  leave  527  ? 

26.  What  is  the  difference  between  487068  and  24703  ? 


k 


46 


ARITHMETIC. 


27.  If  you  divide  3880  dollars  between  two  men,  giving 
one  1907  dollars ;  how  much  will  you  give  the  other  ? 

28.  Subtract  2222  from  3111.    • 

29.  Subtract  9  from  1000. 

30.  Seven  millions,  minus  seventeen,  is  how  much  ? 


V.    DIVISION. 

61  •   The  Principles  of  Division  Illustrated. 

1.  A  butcher  having  35  sheep,  began  Monday  morning, 
and  killed  5  every  morning  as  long  as  they  lasted ;  how  many 
days  did  they  last  ? 

Since  he  killed  5  sheep  each  day,  they  would  last  as  many 
days  as  there  are  times  5  sheep  in  35  sheep. 

After  he  had  killed  5,  Monday,  30  remained ;  Tuesday, 

25  remained ;  Wednesday,  20  remained  ; 

35  Thursday,  15  remained  ;  Friday,  10  re- 

5  Monday.         mained;    Saturday,    5   remained;    and, 

—  Sunday,  he  killed  the  last  5 ;  and  none 
30  remained.     Hence  they  lasted  7  days. 

5  Tuesday.  But  when  it  is  to  be  ascertained  how 

—  many   times   a   given    number    can    be 
25  subtracted  from  another  given  number, 

5  Wednesday,  that  is,  how  many  times  a  subtrahend  is 

—  contained  in  a  minuend,  it  can  be  done 
20  by  a  shorter  process  than  subtracting  once 

5  Thursday,      the  subtrahend  at  a  time. 

—  Write  35,  the  minu- 
15                            5)  35  (7  days.       end  ;  draw  a  line  on 

5  Friday.  35  each  side,   to   distin- 

—  —  guish  it  from  the  other 
10                                 00                     numbers  to  be  written 

5  Saturday.  with  it,  and  at  the  left 

—  hand,  write  5,  the  subtrahend.  Now, 
5  think  how  many  5s  there  are  in  35,  and 
5  Sunday.  place  the  number  at  the  right  hand.     To 

—  ascertain  whether  you  thought  the  right 
0  number,  subtract  so  many  times  5  all  at 

once.  If  there  is  nothing  left,  your  num- 
ber is  right ;  for,  if  there  are  exactly  7  fives  in  35,  then  the 
sum.  of  7  times  5,  subtracted  from  35,  should  leave  nothing. 


DIVISION.  47 

"teacher  having  48  scholars  studying  arithmetic, 
separated  them  into  classes  of  12  scholars  each;  how  many 
dosses  did  he  nt&ke  ? 

Since  he  put  12  scholars  into  each  class,  he  would  make 

as  many  classes  as  there  are  times  12  scholars  in  48  scholars. 

Write  the   48 ;  draw  a  line  on    each 

12) 48 (4  classes,    side;  and  write  the  12  at  the  left  hand. 

48  Now,  how  many  12s  do  you  think  there 

—  are   in   48  ?      Four    12s.      Very   well ! 
00                     Place  the  4  at  the  right  hand,  and  ascer- 
tain whether  4  such  classes  take  exactly 

all  of  the  48  scholars. 

3.  A  butcher  killed  35  sheep  in  7  days ;  how  many  would 
ihat  be  each  day  ? 

Killing  one  each  day  would  require  7  sheep ;  therefore,  he 

would  kill  as  many  each  day,  as  he  had  times  7  sheep. 

Write  the  35,  draw  the  lines,  and  write  the  7  at  the  left 

hand,  think  how  many  7s  there 

7)  35  (5  sheep  a  day.  are  in  35,  and  place  the  number 

35  at  the  right  hand.     This  num- 

—  ber  is  the  answer  required,  if  7 
00                                     multiplied  by  it  make  exactly 

35. 

4.  A  teacher  having  48  scholars  studying  arithmetic, 
separated  them  into  4  equal  classes  ;  how  many  could  he  put 
into  each  class  ? 

Putting  one  into  each  class  would  require  4  scholars ; 
therefore,  he  could  put  as  many  into  each  diss,  as  he 
had  times  4  scholars. 

Arrange     the    tvo    given 

4)48(12  scholars  a  class.       numbers,   think    how   many 

48  4s  there  are  in  48,  ana  place 

—  the  number  at  the  right  hand 
00                                         for     the    answer     required. 

Then    ascertain   whether    4 
multiplied  by  this  number,  take  exactly  all  the  scholars. 

<ft3.    Observation. 

Observe,  in  the  first  and  second  examples,  (Olj)  that 
the  purpose  is  to  divide  a  number  into  equal  parts  of  a  given 
SIZE,  TO  ascertain  the  number  of  such  parts  ;  hut  in  the  third 
arid  fourth  y  that  the  purpose  is  to  divide  a  number  into  a 


48  ARITHMETIC. 

GIVEN  NUMBEE  of  equal  parts,  to  ascertain  the  size  of  such 
parts. 

Observe,  also,  that  each  of  these  purposes  is  effected  by 
ascertaining  how  many  times  one  given  number  is  con- 
tained IN  another. 

63.  Definition  of  Terms,  and  the  Sign  for  Division. 

Division  is  the  separating  of  a  number  into  equal  parts  of  a 
given  size,  or  into  a  given  number  of  equal  parts. 

Dividend  is  a  number  to  be  divided  into  equal  parts  of  a 
given  size,  or  into  a  given  number  of  equal  parts. 

Divisor  is  a  number  which  expresses  either  the  size,  or 
'  number  of  the  equal  parts  to  be  made  of  the  dividend. 

Qicotient  is  the  required  number  which  must  express  either 
the  number,  or  size  of  the  equal  parts  made  of  the  dividend. 

Jr  A  horizontal,  line  between  two  dots,  is  the  sign  for 
division.  It  implies,  that  what  precedes  the  sign  is  to  be 
divided  by  the  number  which  follows  it. 

Thus  ;  35  -7-  5==  7,  which  is  read,  35  divided  by  5  equals 
7 ;  or,  5  in  35,  7  times.  Here  35  is  the  dividend,  5  the 
divisor,  and  7  the  quotient. 

64.  Exercises  for  Illustrating  the  Principles  of  Divi- 

sion. 

Solve  and  explain  the  following  problems  on  the  left,  like 
the  first  and  second  ;  and  those  on  the  right,  like  the  third  and 
fourth  of  the  preceding  examples,  (61.) 

1.  If  a  man  have  15  apples,  I  2.  A  man  gave  15  apples 
to  how  many  boys  could  he  !  equally  to  5  boys  ;  how  many 


p^ive  3  apples  apiece  ? 

3.  How  many  oranges,  at 
6  cents  apiece,  can  you  buy 
for  24  cents  ? 

5.  How  many  apples,  at 
3  cents  apiece,  can  you  buy 
for  18  cents  ? 

7.  How  many  barrels  of 
flour,  at  8  dollars  a  barrel, 
could  you  buy  for  40  dollars  ? 


would  that  be  for  each  boy  ? 

4.  If  you  should  pay  24 
cents  for  4  oranges,  how  much 
would  they  cost  apiece  ? 

-  6.  If  6  apples  cost  18  cents, 
what  is  the  cost  of  each  ap- 
ple ? 

8.  If  you  should  pay  40 
dollars  for  5  barrels  of  flour, 
what  would  be  the  price  of 
each  barrel  ? 


DIVISION. 


49 


9.  If  6  shillings  make  a 
collar,  how  many  dollars  in 
42  shillings  ? 

11.  If  beef  cost  9  cents  a 
pound,  how  much  could  be 
I  ought  for  54  cents  ? 

13.  Mr.  Jones  bought  sugar 
at  7  cents  a  pound, expending 
49  cents ;  how  many  pounds 
c  id  he  get  ? 

15.  How  many  pews  would 
eccommodate  63  persons,  if 
7  persons  could*  sit  in  one 
pew? 

17.  If  8  ninepences  make 
cne  dollar,  how  many  dollars 
in  72  ninepences  ? 

19.  How  many  classes,  of 
10  scholars  each,  in  a  school 
c  f  80  scholars  ? 

21.  How  many  sections 
could  be  made  in  a  company 
of  64  soldiers,  if  8  soldiers 
make  a  section  ? 

23.  How  many  3s  are  there 
in  12? 

25.  How  many  3s  can  be 
subtracted  from  15? 

27.  How  many  times  can 
3  be  subtracted  from  IS  ? 

29.  How  many  times  is  3 
contained  in  21  ? 

31.  How  many  times  4 
equal  20  ? 

33.  Into  how  many  parts 
of  4  each,  can  24  be  sepa- 
rated ? 

35.  Into  how  many  parts 
of  10  each,  can  80  be  sepa- 
rated ? 

37.  Into  how  many  parts 
of  6  each  can  35  be  divided  ? 


10.  How  many  shillings  in 
a  dollar,  if  42  shillings  make 
7  dollars  ? 

12.  What  would  be  the 
cost  of  1  pound  of  beef,  if 
6  pounds  cost  54  cents  ? 

14.  If  Mr.  Jones  should 
expend  49  cents  for  7  pounds 
of  sugar,  how  much  would 
that  be  a  pound  ? 

16.  If  63  persons  would 
fill  9  pews,  how  many  per- 
sons would  be  accommodated 
in  one  pew  ? 

18.  If  72  ninepences  make 
9  dollars,  how  many  nine- 
pences make  one  dollar? 

20.  If  80  scholars  be  put 
into  8  equal  classes,  how  large 
would  be  the  classes  ? 

22.  Make  8  equal  sections 
of  64  soldiers,  and  tell  me 
how  many  soldiers  you  put 
into  a  section  ? 

24.  Four  times  what  num- 
ber makes  12  ? 

26.  Five  times  what  num- 
ber will  amount  to  15  ? 

28.  What  number  taken  6 
times  will  equal  18? 

30.  What  number  taken  7 
times  makes  21  ? 

32.  Five  times  what  num 
ber  equals  20  ? 

34.  If  24  be  separated  into 
6  equal  parts,  how  many  in 
each  part  ? 

36.  If  30  be  separated  into 
3  equal  parts,  how  large  is 
each  part  ? 

38.  If  35  be  divided  into  7 
equal  parts,  how  large  is  each 
part? 


60 


ARITHMETIC. 


39.  Into  how  many  parts 
01  5  each  can  45  be  divided  ? 

41.  What  number  must  6 
be  multiplied  by  to  make  48  ? 

43.  What  number  must  7 
be  multiplied  by  to  make  56  ? 

45.  Divide  63  into  equal 
parts  of  7  each.  How  many 
are  the  parts  ? 

65.    Division  Table. 

In  order  to  perform  Division  with  facility^  yoa  will,  before 
attempting  further  progress,  correctly  ascertain,  and  thoroughly 
commit  to  memory  the  quotient  of  each  combination  of  two 
numbers  in  the  following  table. 


40.  If  45  be  divided  into  9 
equal  parts,  how  large  is  each 
part  ? 

42.  What  number  multi- 
plied by  8  will  make  48  ? 

44.  What  number  multi- 
plied by  8  will  make  56  ? 

46.  Divide  63  into  9  equal 
parts.  How  large  are  the 
parts  ? 


2-^2  = 

3- 

r-  3  = 

4^4  = 

5- 

■-5 

44-2  = 

6-3  = 

8-^4  = 

10- 

r-5 

6.~2  = 

9-^3  = 

12-7-4  = 

15- 

'-d 

8-r-2  = 

12h-3  = 

16-f-4  = 

20- 

-5 

10-^2  = 

15-7-3  = 

20-7-4  = 

25- 

-5 

12-^2  = 

18^3  = 

24-7-4  = 

30- 

-5 

14 -f- 2  = 

21^3  = 

28-7-4  = 

35- 

-5 

16-^2  = 

24-^3  = 

32 -T- 4  = 

40- 

r5 

18 -T- 2  = 

27-7-3  = 

36-^4  = 

45- 

r-5 

6-r-6  = 

7- 

-7  = 

8^8  = 

9- 

-9 

12 --6  = 

14- 

-7  = 

l6-^8  = 

18- 

-9 

18 -f- 6  = 

21  - 

-7:=: 

24-^8  = 

27-7 

-9 

24  H- 6  = 

28- 

—  7  — 

32 -r- 8  = 

36  H 

-9 

30-7-6  = 

35- 

-7  = 

40-7-8  = 

45-- 

-9 

36-7-6  = 

42- 

—  7  — 

48 -r- 8  = 

54  H 

-9 

42-7-6  = 

49- 

-7  = 

56-^8  = 

63  H 

-9 

48-^6  = 

56- 

-7  = 

64^8  = 

72-7 

-9 

54-^6  = 

63- 

-  7  = 

72-7-8  = 

81-; 

-9 

u   Model  of 

A    Rl 

£CITATIO.> 

r. 

66. 

1.    How  many  quarts  are  there  in  600  pints? 
Since  there  are  2  pints  in  a  quart,  there  will  be  as  many 
quarts  as  there  are  times  2  pints  in  600  pints. 


DIVISION.  ,  61 

2  is  contained  3  times  in  6  units  of  the  first  order,  but,  in 

6  units  of  the  third  order,  which 

2 )  600  ( 300  quarts.         are  100  times  as   large,    (6,)   it 

600  must   be  contained  100  times  as 

often,  which  is  300  times.     300 

quarts,  at  2  pints  each,  take  600 
)ints,  which  subtracted  from  600  pints,  nothing  remains, 
rlence,  300  quarts  is  the  answer  required. 

^Sy.   Exercises  in  Dividing  Units  of  any  one  Order. 

In  like  manner,  solve  and  explain  the  following  prob- 
lems, 

1.  If  in  a  certain  school-room  2  scholars  sit  at  a  desk, 
!iow  many  desks  will  accommodate  200  scholars  ? 

2.  If  in  a  certain  school  there  are  80  scholars,  and  2 
;eachers,  how  many  scholars  are  there  for  each  teacher  ? 

3.  If  a  man  pay  3  dollars  apiece  for  hats,  how  many  hats 
can  he  buy  for  90  dollars  ? 

4.  If  Mr.  Farmer  sell  2  cows  for  40  dollars,  how  much  is 
i:hat  apiece  ? 

5.  At  4  dollars  a  yard  for  cloth,  how  many  yards  can  be 
bought  for  80  dollars  ? 

6.  If  800  dollars  a  year  be  paid  to  4  female  teachers,  hoi?^ 
much  is  that  apiece  ? 

7.  At  the  rate  of  5  miles  an  hour,  how  long  would  it  take 
to  travel  500  miles  ? 

8.  If  6  shares  in  a  bank  cost  600  dollars,  how  much  is 
that  a  share  ? 

9.  At  an  average  of  7  persons  to  a  family,  how  many 
families  in  a  town  of  7000  persons  ? 

10.  If  90000  dollars  be  the  cost  of  3  miles  of  rail-road, 
what  is  the  cost  per  mile  ? 

68.   Model  of  a  Recitation. 

1.  A  hatter  made  in  a  year  560  hats,  and  packed  them  foi 
market  in  boxes  holding  8  hats  apiece.  How  many  boxes 
would  he  need  ? 

Since  each  box  would  hold  8  hats,  he  would  need  as  many 
boxes  as  there  are  'times  8  hats  in  560  hats.     But,  one  unit 


GZ  ARITHMETIC. 

of  any  order  making  ten  units  of  the  next  lower  order,  (10) 

the  5  hundreds  are  equal  to  50  tens, 

8  )  560  (  70  boxes.         which  with  the  6  tens,  make  56 

560  tens ;   8  is  contained  7  times  in  56 

units  of  the  first  order,  but  in  56 

units  of  the  second  order,  which 
are  10  times  as  large,  (Aj)  it  must  be  contained  10  times  as 
often,  which  is  70  times ;  70  boxes  at  8  hats  each,  would 
take  560  hats,  which  subtracted  from  560  hats,  nothing 
remains.     Hence,  70  boxes  is  the  answer  required. 

69*   Exercises    in    Reducing    Units    of   a  High   to   a 
Lower  Order  for  Division. 

In  like  manner^  solve  and  explain  the  following  prob' 
lems, 

1.  How  many  pairs  of  boots  could  be  bought  for  150  dol- 
lars at  3  dollars  a  pair  ? 

2.  If  350  dollars  be  paid  for  5  horses,  how  much  is  that 
apiece  ? 

3.  How  many  hours  will  it  take  to  travel  350  miles  at  7 
miles  per  hour  ? 

4.  If  a  stage  travel  120  miles  in  12  hours,  how  far  is  that 
an  hour  ? 

5.  How  many  times  is  5  contained  in  450  ? 

6.  Into  how  many  parts  of  9  each  can  6300  be  divided  ? 

7.  If  3200  be  divided  into  8  equal  parts,  how  large  are  the 
parts  ? 

8.  Divide  2500  into  5  parts  ;  how  large  is  each  part  ? 

9.  What   number   must  7  be  multiplied  by  to   produce 
4900  ? 

10.  What  number  multiplied  by  3  will  produce  27000  ? 

11.  Divide  100  by  4. 

12.  Divide  1000  by  8. 

13.  If  ISOO  be  the  dividend  and  9  the  divisor,  what  will 
be  the  quotient  ? 

TO.  Explanation  of  the  Written  Process  of  Division. 

1.    How  many  yards  are  there  in  9636  feet  ? 
Since  there  are  3  feet  in  a  yard,  there  will  be  as  many 
yards  as  there  are  times  3  feet  in  9636  feet. 


■' 


DIVISION. 


S3 


3  )  9636  (  3000  yards. 

200  yards. 

10  yards. 

2  yards. 

3212  yards. 


9000 

636 
600 

36 
30 

6 
6 


3  is  contained  3  times  in  9  units  of  the  first  order,  but  in 
9  units  of  the  fourth  order,  it 
must  be  contained  1000  times 
as  often,  (lO,)  that  is,  3000 
times;  3000  yards  at  3  feet 
each,  take  9000  feet,  which  sub- 
tracted from  9636  feet,  leave 
636  feet;  3  is  contained  in  6 
2mits  2  times ;  therefore,  in  6 
hundreds,  it  is  contained  200 
times ;  200  yards  at  3  feet  each 
take  600  feet,  which  subtracted 
from  636  feet  leave  36  feet ; 
3  is  contained  in  3  U7iiis  1  time, 
therefore,  in  3  tens  it  is  con- 
tained 10  times;  10  yards  at  3  feet  each  take  30  feet,  which 
subtracted  from  36  feet  leave  6  feet,  in  which  3  is  contained 
2  times ;  2  yards  at  3  feet  each  take  6  feet,  which  subtracted 
from  6  feet,  nothing  remains.  Hence,  3000  yards  -|-  200 
}ards  -j-  10  yards  -}-  2  yards  =  3212  yards,  is  the  answer 
required. 

This  operation  may  be  abridged  by  omitting  some  unnec- 
essary figures.  Instead  of  the 
ciphers  belonging  to  the  first 
number  in  the  quotient,  write 
the  digits  of  the  other  numbers 
as  they  are  obtained,  which  will 
finally  leave  each  figure  in  its 
own  place. 

The  product  of  the  divisor 
and  the  first  quotient  figure  is  9 
thousand  ;  omitting  the  ciphers, 
it  will  be  sufficient  to  write  the 
9  in  the  thousands'  place,  and 
subtract  it  from  the  thousands ; 
then  bring  down  the-  6  hundreds 
only,  for  consideration  ;  200  times  the  divisor  is  6  hundreds, 
which  being  subtracted  from  the  hundreds,  bring  down  the  3 
tens ;  10  times  the  divisor  is  3  tens,  wMch  being  subtracted 
from  the  tens,  bring  down  the  6  units ;  2  times  the  divisor  is 
6  units,  which  being  subtracted  from  the  units,  nothing  more 
5* 


3  )  9636  (  3212  yards. 
9-- 

6- 
6- 

3- 
3- 

6 
6 


54  ARITHMETIC. 

of  the  dividend  remains.     Hence,  3212  yards  is  the  answer 
required,  as  before. 

71,  Model  of  a  Recitation. 

Divide  2848  by  4,  or  find  how  many  times  4  is  contained 
in  2848. 

4  is  contained  7  times  in  28  units,  but  in  28  hundreds  it  is 

contained  100  times  as  often,  (68^)  or 

4  )2848{  712         7  hundred  times;    7  hundred  times  4 

28  are  28  hundred,  which    subtract  from 

the  hundreds,  and   bring  down   the   4 

4  tens ;  4  is  contained  1  time  in  4  units,  but 

4  in  4  tens  it  is  contained  10  times  as 

— •  often,  or  1  ten  times  ;    10  times  4  are 

8  4  tens,  which  subtract  from  the  tens  and 

8  bring  down  the  8  units ;    4  is  contained 

—  2  times  in  8  units  ;  2  times  4  are  8,  which 

subtracted,    nothing    remains ;    conse- 
quently, 712  is  the  result  required. 

72.  Exercises  in  Explaining  the  Written  Process  of 

Division. 

In  like  manner,  solve  and  explain  the  follounng  probhrns, 

1.  How  many  bushels  in  88  pecks  ? 

2.  How  many  weeks  in  77  days  ? 

3.  How  many  dollars  in  126  shillings  ? 

4.  If  4  horses  are  required  to  draw  1  wagon,  how  many 
wagons  might  be  drawn  by  168  horses  ? 

5.  If  a  man  can  travel  5  miles  an  hour,  how  many  hours 
would  it  take  him  to  travel  205  miles  ? 

6.  A  drover  received  248  dollars  for  sheep  that  he  sold  for 
4  dollars  a  head.     How  many  were  there  ? 

7.  If  5  bushels  of  corn  pay  for  a  pair  of  boots,  how  many 
pairs  would  255  bushels  pay  for  ? 

8.  Suppose  6  men  should  contribute  186  dollars,  how 
much  would  that  be  apiece  ? 

9.  Suppose  355  dollars'  bounty  were  paid  at  5  dollars 
apiece  to  a  company  of  soldiers.  How  many  soldiers  in  the 
company. 

10.  How  many  weeks  can  a  man  get  board  for  156  del 
lars,  at  3  dollars  a  week  ? 

11.  How  many  times  is  7  contained  in  637? 


mm-. 


DIVISION^  66 


Stippose  3699  to  be  a  dividend,  and  9  a  divisor,  what 
is  the  quotient? 

13.  Divide  1S36  by  3. 

14.  What  must  I  multiply  by  8  to  make  7288  ? 

15.  Into  how  many  parts  of  5  each  can  555  be  divided  ? 

16.  If  567  be  divided  into  7  equal  parts,  what  must  be  the 
size  of  each  part? 

73.  Model  of  a  Recitation. 

Mr.  Farmer  planted  4785  grains  of  corn  in  a  field,  planting 
I'i  grains  in  each  hill.     How  many  hills  did  he  make  ? 

Since  he  put  5  grains  in  each  hill,  he  made  as  many  hills 
IS  there  are  times  5  grains  in  4785  grains. 

Beginning  at  the  left  hand  of  the  dividend,  take  into  con- 
sideration the  fewest   figures  that 
5 )  4785  ( 957  hills.         can  contain  the  divisor ;  as  5  is  not 
45  contained  in  4,  take  47  hundreds, 

in  45  of  which  5  is  contained  9 

28  hundreds    times,  (lO,)  900   hills 

25  require  45  hundred  grains,  which 

subtracted  from  47  hundred  leave 

35  2  hundred,  with  which  join  the  8 

35  ^       tens,  making  28  tens,  (IO5)  in  25 

—  of  which  5  is  contained  5  tens  times ; 

50  hills   require   25   tens    grains, 

which  subtracted  from  28  tens  leave  3  tens,  with  which  join 

the  5  units,  making  35  units,  in  which  5  is  contained  7  times  ; 

7  hills  require  35  grains,  which  subtracted  from  35  grains, 

nothing  remains.     Hence,  957  hills  is  the  answer  required. 

74.  Observation. 

Observe,  (73^)  that  the  division  is  comm€?iced,  by  dividing 
the  fewest  figures  on  the  left  of  the  dividend  that  will  contain 
the  divisor,  that  the  quotient  figure  will  be  of  the  same 
denomination  as  that  part  of  the  dividend  from  which  it  is 
obtained^  that  each  succeeding  figure  of  the  dividend  will 
require  an  additional  figure  in  the  quotient,  a  cipher  if 
nothing  larger,  that  the  products  of  the  divisor,  by  each 
quotient  figure,  are  to  be  subtracted  from  those  parts  of  the 
dividend  from  lohich  the  respective  quotient  figures  are 
obtained,  that  the  remainder  in  each  case  is  reduced  (208) 
and  united  to  the  units  of  the  next  lower  order ^  for  division^ 


56 


ARITHMETIC. 


and  that  the  sum  of  these  partial  products^  or  the  product  of 
the  divisor  by  the  whole  quotient^  is  equal  to  the  dividend. 

75.  Proof  of  Division. 

To  prove  the  correctness  of  an  operation  in  Division,  multi- 
ply the  divisor  and  quotient  together  ;  if  their  product  equals 
the  dividend,  probably  the  operation  is  correct ;  for,  the  cor- 
rect quotiem,  expressing  how  many  times  the  divisor  there 
are  in  the  dividend,  (61,)  is  one,  and  the  divisor  the  other 
of  two  factors,  whose  product  should  be  the  dividend. 

76.  Exercises  requiring  some  Units  of  each  Order  to 
BE  reduced  to  a  Lower  Order  for  Division. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  If  9  hills  of  potatoes  yield  a  bushel,  how  many  bushels 
of  potatoes  in  a  field  of  1296  hills  ? 

2.  If  an  army  of  2048  men  were  marching  in  sections, 
having  8  men  in  each  section,  how  many  sections  would  be 
there  ? 

3.  If  in  an  army  every  ninth  man  is  an  officer,  how  many 
officers  in  an  army  of  4608  men  ? 

4.  If  a  general  should  divide  his  army  of  12096  men  into 
7  equal  divisions,  how  many  men  would  be  in  each  division  ? 

5.  How  many  weeks  in  364  days  ? 

6.  How  many  Sabbath  days  in  12852  days  ? 

7.  If  an  acre  of  land  pasture  5  sheep,  how  many  acres 
could  pasture  315  sheep  ? 

8.  How  many  times  is  6  contained  in  738  ? 

9.  How  many  times  is  4  contained  in  20012  ? 

10.  Divide  3606  by  3. 

11.  Divide  25634  by  2. 

12.  If  28028  be  a  dividend,  and  7  a  divisor,  what  is  the 
quotient  ? 

13.  If  18675  be  a  product,  and  5  one  factor,  what  is  the 
other  factor  ? 

14.  What  must  11889  be  divided  by,  to  give  9  for  a  quo- 
tient ? 

15.  "What  must  8  be  multiplied  by,  to  produce  2496  ? 

77.  Model  of  a  Recitation. 

1.  If  in  the  month  of  July  a  rail-road  company  received 


DIVISION.  57 

6284  dollars  from  passengers,  at  2  dollars  apiece,  how  many 
passengers  rode  in  the  cars  in  that  month  ? 

Since  each  passenger  paid  2  dollars,  there  were  as  many 
passengers  as  there  are  times  2  dollars  in  6284  dollars. 

To  obtain  the  answer  by  a  still  shorter  process,  write  the 
cividend  and  divisor  as  heretofore,  but  perform  the  operation 
ii  your  mind,  writing  only  the  quotient,  and  write  that  under 
the  dividend,  with  each  figure  under  that  of  its  own  order. 

Thus,  2  in  6  thousands  3  thousand 
2)  6284  times,  therefore,  write  3  in  the  thou- 

sands'  place  ;   2  in  2  hundreds  1 

3142  passengers.         hundred  times,  therefore,  write  1  in 

the  hundreds'  place ;  2  in  8  tens  4 
tsns  times,  therefore,  write  4  in  the  tens'  place  ;  2  in  4  units 
fil  times,  therefore,  write  2  in  the  units'  place  :  making  3142 
times  2  dollars.  Hence,  3142  passengers  is  the  answer  re- 
quired. 

2.  If  a  stage  run  6  miles  an  hour,  how  many  hours  would 
i^.  take  the  stage  to  run  1848  miles  ? 

Since  in  one  hour  it  runs  6  miles,  it  will  take  as  many 
hours  *as  there  are  times  6  miles  in  1848  miles. 

6  in  18  hundreds  3  hundreds  times ;  write 
6 )  1848  3  in  the  hundreds'  place.    If  6  were  con- 

tained  in  the  4,  which  is  tens,  the  quo- 

308  hours.         tient  figure  would  be  tens,  but  as  6  is 

not  contained  in  4,  there  are  no  tens  in 
the  quotient,  therefore,  write  a  cipher  in  the  tens'  place,  and 
reduce  the  4  tens  to  units,  making  40  units,  which,  joined 
with  the  8  units,  make  48  units,  in  which  6  is  contained  8 
times,  therefore,  write  8  in  the  units'  place  :  making  308 
times  6.     Hence,  308  hours  is  the  answer  required. 

78.   Exercises  in  Abridging  the  Process  of  Division. 

In  like  jnanner,  solve  and  explain  the  following  problems. 

1.  If  306  dollars  be  divided  among  3  men,  what  is  each 
man's  share  ? 

2.  If  4  shares  of  a  bank  cost  416  dollars,  what  would  one 
share  cost  ? 

3.  If  six  brothers  receive  a  legacy  of  1512  dollars,  what 
would  be  the  share  of  each  ? 

4.  Paid  150  dollars  for  6  tons  of  hay.  How  much  was 
that  for  a  ton  ? 


I 


53  ARITHMETIC. 

5.  If  there  are  1280  inhabitants  in  a  town,  and  the  families 
average  8  persons  apiece,  how  many  families  in  that  town  ? 

6.  How  many  yards  of  cloth  can  be  bought  for  1155  dol- 
lars, at  7  dollars  a  yard  ? 

7.  Find  a  nmnber,  which,  multiplied  by  9,  will  produce 
63234. 

8.  What  number,  multiplied  by  8,  will  produce  2464  ? 

9.  What  number,  divided  by  9,  will  give  72  for  a  quotient  ? 

10.  If  7  be  a  divisor,  and  42014  a  dividend,  what  is  the 
quotient  ? 

11.  How  many  times  is  5  contained  in  1204500890  ? 

12.  How  many  times  does  540010  contain  5  ? 

13.  How  many  times  8  are  there  in  25648  ? 

14.  Divide  4004  by  4. 

15.  Divide  16800  by  8. 

16.  Divide  36900  by  3.  'WtH^ 

17.  Divide  1800108  by  9.  WW 

18.  Divide  105105  by  7. 

19.  If  1836  be  a  dividend,  and  9  the  divisor,  what  is  the 
quotient  ? 

20.  If  1728  be  divided  by  9,  what  would  be  the  quotient  ? 

21.  If  72  be  a  dividend,  and  9  the  quotient,  what  is  the 
divisor  ? 

22.  If  63  be  a  dividend,  and  7  the  quotient,  what  is  the 
divisor  ? 

79 •  Model  of  a  Recitation. 

1.  How  many  days  in  1728  hours  ? 

Since  in  one  day  there  are  24  hours,  there  must  be  as 
many  days  as  there  are  times  24  hours  in  1728  hours. 

24  is  contained  in  172  tens  7  tens 

24)  1728  (72  days.         times  ;  70  times  24  make  168  tens, 

168  which,  subtracted  from  172  tens,  leave 

4  tens,  to  which  bring  down  the  8 

48  units,  making  48  units,  in  which  24 

48  is  contained  2  times  ;    2  times  24 

—  make  48,  which  subtracted  from  48, 

nothing  remains.     Hence,  as    there 

are  72  times  24  hoiirs,  72  days  is  the  answer  required. 

2.  How  many  times  is  64237  contained  in  436940074  ? 
The  many  figures  in  thfs  divisor,  present  a  difficulty  in 

ascertaining  any  quotient  figure.     The  best  way  is  to  seek 


Divisioif.  59 

hjw  many  times  the  highest  figure  only,  of  the  divisor,  is 
c«mtained  in  the  highest  one,  or  two,  figures  of  the  dividend ; 
this  quotient  figure  will  either  be  right,  or  one  or  two  too 
krge  ;  for  the  greater  certainty,  however,  before  multiplying 
the  whole  divisor  by  it,  multiply  mentally  only  one  or  two  of 
the  highest  figures  of  the  divisor,  and  compare  the  product 
v^ith  the  highest  figures  of  the  dividend  from  which  this  part 
o  ■  the  product  is  to  be  subtracted  ;  if  the  appearance  is  satis- 
fy-ctory,  proceed  with  this  quotient  figure,  otherwise  take  a 
smaller  figure,  and  proceed. 

If  at  any  time  a  product  prove  too  large  to  be  subtracted, 
the  last  quotient  figure  is  too  large  ;  or,  if  a  remainder  be 
k.rger  than  the  divisor,  the  last  quotient  figure  is  too  small. 
Ill  either  case,  erase  it,  and  try  another  figure. 

6  is  contained  7  times  in 

64237)  436940074  (6802  times.       43,  but    7   times    64    is 

385422  greater  than  436;  there- 

fore,  7  is  too  large  for  the 

515180  first  quotient  figure  ;  write 
513896  6    in    the    quotient,  and 
■  subtract  6  thousand  times 
128474                            the  divisor,  that  is,  6  times 
128474                            the  divisor  from  the  thou- 
sands,  and  to  the  remain- 
der bring  down  the  next 
figure  of  the  dividend ;  6  is  contained  8  times  in  51,  and  8 
times  64  being  less  than  515,  subtract  8  hundred  times  the 
divisor,"  that  is,  8  times  the  divisor  from  these  hundreds,  and 
to  the  remainder  bring  down  the  next  figure  ;  this  number 
being  smaller  than  the  divisor,  there  can  be  no  tens  in  the 
quotient;  therefore,  write  a  cipher  in  the  tens'  place,  (77'^) 
and  bring  down  the  next  figure  ;  6  is  contained  in,  12  twice  ; 
subtract  2  times  the  divisor,  and  nothing  remains.     Hence, 
6802  times,  is  the  answer  required. 

80,   General  Exercises  in  Division. 

In  like  manner,  solve  and  'explain  the  following  problems, 

1.  How  many  days  in  360  hours  ? 

2.  If  a  man  travel  45  miles  a  day,  in  how  many  days  will 
he  travel  1125  miles  ? 

3.  A  butcher  gave  875  dollars  for  35  cows.    What  was  the 
t  of  each  cow  ? 


f 


M 


ARITHMETIC. 


4.  If  a  field  of  34  acres  produce  1020  bushels  of  corn,  how 
much  would  that  be  per  acre  ? 

5.  Suppose  an  acre  of  land  to  produce  38  bushels  of  corn, 
how  many  acres  must  be  cultivated  to  produce  4902  bushels  ? 

6.  How  many  horses,  at  75  dollars  apiece,  can  be  bought 
for  1125  dollars? 

7.  A  school-district  paid  a  teacher  144  dollars  for  teaching, 
at  36  dollars  a  month.     How  long  was  the  school  kept  ? 

8.  If  a  man's  income  be  1095  dollars  for  365  days,  how 
much  is  that  per  day  ? 

9.  How  many  hogsheads,  of  63  gallons  each,  can  be  filled 
from  8379  gallons  ? 

10.  How  many  years  in  8395  days,  if  365  days  be  called 
a  year  ? 

11.  If  1512  dollars  be  divided  among  some  brothers,  so 
that  each  may  receive  252  dollars,  how  many  are  the  brothers  ? 

12.  How  many  bank  shares  can  be  purchased  with  2912 
dollars,  at  112  dollars  each  ? 

13.  How  many  acres  of  land  will  yield  6996  bushels  of 
potatoes,  if  212  bushels  grow  on  one  acre  ? 

14.  How  many  barrels  must  a  man  have  to  fill  from  125440 
pounds  of  flour,  if  each  barrel  hold  196  pounds  ? 

15.  A  man  put  17484  pounds  of  tea  into  186  chests.  How 
much  in  each  chest  ? 

16.  How  many  times  can  48  be  subtracted  from  5040  ? 

17.  How  many  times  is  75  contained  in  23025  ? 

18.  How  many  times  25  is  equal  to  23025  ? 

19.  How  many  times  does  105735  contain  105  ? 

20.  How  many  times  does  105735  contain  1007  ? 

21.  Divide  144144  into  144  equal  parts  ;  what  is  each 
part  ? 

22.  Divide  172800  nuts  among  some  boys,  giving  them 
1440  nuts  apiece.  How  many  boys  can  you  supply  with 
them  ? 

23.  What  number,  multiplied  by  754,  will  produce  18850  ? 

24.  The  product  of  two  factors  is  612060.  If  one  factor 
is  303,  what  is  the  other  factor  ?  • 

25.  Divide  a  city  of  78612  inhabitants  into  12  equal  wards. 
How  many  inhabitants  in  each  ward  ? 

26.  How  many  equal  parts  can  be  made  of  1048576,  if 
1024  be  one  of  the  parts  ? 

27.  How  many  times  409^  is  equal  to  262144  ? 


DIVISION.  61 

28.  If  2048  be  one  of  a  certain  number  of  equal  parts  of 
^  31 072,  how  many  are  the  parts  ? 

\it.   General  Exercises  in  the  Fundamental  Principles 
OF  Arithmetic. 

1.  There  are  two  numbers,  of  which  the  greater  is  27 
times  the  less,  and  the  less  is  contained  9  times  in  27.  What 
are  the  numbers  ? 

2.  A  was  born  when  B  was  26  years  old.  How  old  will 
A  be  when  B  is  45  ? 

3.  If  the  sum  of  3  numbers  be  500,  the  difference  between 
tlie  least  and  the  greatest  be  174,  and  the  difference  between 
tlie  middle  number  and  the  sum  of  the  3  numbers  be  350, 
what  are  the  numbers  ? 

4.  A  man  bought  5  pieces  of  cloth  at  44  dollars  each,  974 
piirs  of  shoes  at  2  dollars  a  pair,  600  pieces  of  calico  at  6 
dDllars  each,  and  sold  the  whole  for  6000  dollars.  How 
niuch  did  he  gain,  or  lose  ? 

5.  A  man  exchanged  6  cows  at  15  dollars  each,  a  yoke  of 
oxen  at  67  dollars,  for  a  horse  at  50  dollars,  and  a  chaise. 
What  did  the  chaise  cost  ? 

6.  A  boy  bought  some  apples,  and,  after  giving  away  10, 
and  buying  34  more,  he  divided  half  of  what  he  then  had 
among  4  companions,  giving  them  8  apiece.  How  many 
apples  did  he  buy  at  first  ? 

7.  What  is  that  number,  to  which,  if  4  be  added,  from 
which  7  be  subtracted,  the  remainder  multiplied  by  8,  and 
the  product  divided  by  3,  the  quotient  will  be  64  ? 

8.  A  man  bought  a  farm  at  25  dollars  an  acre,  and  sold 
half  of  it,  at  the  same  rate,  for  1850  dollars.  How  many 
acres  did  he  buy  ? 

9.  Five  men  and  three  boys  were  paid  a  sum  of  money,  so 
large  that  each  man  had  43  dollars,  and  each  boy  25  dollars. 
What  was  the  whole  sum  ? 

10.  If  a  trader  gain  160  dollars  on  544  barrels  of  flour, 
that  cost  him  6  dollars  a  barrel,  besides  25  dollars  that  he 
paid  for  storage  ;  what  would  he  receive  for  the  flour? 

11.  Suppose  5  bushels  of  wheat  make  a  barrel  of  flour, 
how  many  barrels  can  be  made  from  the  wheat  raised  on  75 
acres,  at  29  bushels  per  acre  ? 

12.  How  many  times  6  in  75  tjmes  29  ? 

13.  A  farmer  exchanges  44  acres  of  land,  worth  36  dollars 
6 


I 


62  ARITHMETIC. 

an  aero,  for  66  acres  of  land  in  another  place.     What  does 
his  land  cost  him  per  acre  ? 

14.  A  man  who  owned  520  acres,  bought  375  acres  more, 
and,  reserving  95  acres  for  himself,  divided  the  remainder 
into  8  equal  farms,  and  sold  them  for  2500  dollars  apiece. 
How  much  did  he  get  per  acre  for  his  land  ? 

15.  If  a  man's  income  be  1349  dollars  a  year,  and  his  ex- 
penses 20  dollars  a  week,  how  much  would  he  save  in  a 
year? 

16.  A  merchant's  business  brought  him,  in  a  year,  2500 
dollars  ;  but  his  expenses  were  1772  dollars.  How  much 
did  he  save  per  week  ? 

17.  If  I  buy  245  hogsheads  of  molasses,  at  18  dollars  each, 
how  much  do  I  gain,  or  lose,  in  selling  it  for  4000  dollars  ? 

18.  If  a  man's  expenses  be  2  dollars  a  day,  and  his  income 
17  dollars  a  week,  how  many  weeks  will  it  take  him  to  save 
156  dollars  ? 

19.  If  a  lot  of  land  be  divided  into  8  farms,  each  of  150 
acres,  and  the  farms  be  sold  for  3000  dollars  apiece,  what 
would  one  acre  cost  ? 

20.  A  gentleman  bought  2  pieces  of  land,  one  contained 
96  acres,  the  other  103  acres.  If  he  should  sell  47  acres,  at 
25  dollars  an  acre,  how  much  would  the  rest  of  the  land  be 
worth  at  the  same  rate  ? 

21.  A  merchant  bought  a  cask  of  molasses  containing  119 
gallons,  and  sold  to  one  man  10  gallons,  to  another  9  gallons, 
to  another  25  gallons.  How  much  is  the  remainder  worth, 
at  40  cents  a  gallon  ? 

22.  What  is  the  difference  between  17  times  105  and  3417 
divided  by  17  ? 

23.  What  is  the  difference  between  20  times  210  and  7 
times  2500  divided  by  175  ? 

24.  If  I  purchase  1200  pounds  of  butter  for  15600  cents, 
how  must  I  sell  it  per  pound  to  gain  2400  cents  ? 

25.  If  I  buy  375  pounds  of  pork  at  7  cents  a  pound,  and 
sell  it  for  3000  cents,  how  much  do  I  gain  on  a  pound  ? 

26.  How  many  quintals  of  fish,  at  2  dollars  each,  will  pay 
for  500  hogsheads  of  salt,  at  5  dollars  a  hogshead  ? 

27.  How  much  flour,  at  7  dollars  a  barrel,  will  pay  for  224 
cords  of  wood,  at  8  dollars  a  cord  ? 

28.  How  many  days  must  3  brothers  work  to  receive  2475 


DIVISION.  63 

celits,  if  one  earn  42  cents  a  day,  the  second  32  cents,  and 
;he  youngest  25  cents  ? 

29.  If  a  man  earn  6  dollars  a  week,  and  his  two  boys  earn 
3  dollars  apiece  a  week,  how  many  weeks  will  it  take  them 
ill  to  earn  624  dollars  ? 

30.  If  a  hogshead  hold  252  quarts,  and  two  boys  work 
:ogether  to  fill  it  with  water,  one  having  a  pail  which  holds 
12  quarts,  the  other  having  a  pail  which  holds  9  quarts,  how 
nany  times  must  they  empty  their  pails  to  fill  the  hogshead  ? 

31.  If  a  full  hogshead  should  begin  to  leak  in  3  places,  at 
3nce,  from  one  hole  4  quarts  a  day,  from  another  2  quarts  a 
day,  and  from  the  other  1  quart  a  day,  how  many  days  before 
the  hogshead  would  be  emptied  ? 

32.  A  man  bought  some  sheep  and  calves,  and  of  each  an 
equal  number,  for  165  dollars,  giving  for  the  sheep  7  dollars 
apiece,  and  for  the  calves  4  dollars  apiece.  How  many  were 
there  of  each  sort  ? 

33.  How  many  coats,  pantaloons  and  vests,  of  each  an 
equal  number,  can  be  made  from  405  yards,  if  it  take  5  yards 
for  a  coat,  3  yards  for  a  pair  of  pantaloons,  and  1  yard  for  a 
vest  ? 

34.  If  9000  men  march  in  a  column  of  750  deep,  how 
many  inarch  abreast  ? 

35.  A  man  left  his  estate,  valued  at  8956  dollars,  to  his 
wife  and  daughters,  giving  his  wife  4688  dollars,  and  his 
daughters  1067  dollars  apiece.    How  many  daughters  had  he  ? 

36.  The  factors  of  a  certain  number  are  the  diiTerence  be- 
tween 1632  and  1700,  and  between  94  and  5  dozen.  What 
is  that  number  ? 

37.  Paid  57600  cents  ^r  eggs,  at  12  cents  a  dozen.  How 
many  eggs  did  I  buy  ? 

38.  A  boy  bought  a  sled  for  96  cents,  exchanged  it  for  8 
quarts  of  nuts,  sold  half  of  his  nuts  at  12  cents  a  quart,  and 
gave  the  rest  of  his  nuts  for  a  penknife,  which  he  sold  for  34 
cents.     How  much  did  he  gain,  or  lose  ? 

39.  Three  men  owned  farms  situated  together ;  the  first  had 
64  acres,  the  second  had  20  acres  more  than  the  first,  and 
the  third  had  as  many  acres  as  both  the  first  and  second ;  the 
three  farms  were  worth  7400  dollars.    What  is  that  per  acre  ? 

40.  If  a  man  owe  728  dollars  to  Mr.  Saveall,  and  works 
for  him  to  pay  the  debt ;  how  many  years,  of  52  weeks  each, 
will  it  take  him,  if  he  pay  only  one  dollar  a  week  ? 


64  ARITHMETIC. 

41.  If  a  man  earn  40  dollars  a  month,  and  spend  13  dol- 
lars of  it  each  month,  how  long  will  it  take  him  to  pay  for 
a  house  worth  1620  dollars  ? 

42.  A  farmer  sold  some  pork  at  17  dollars  a  barrel  to  the 
amount  of  510  dollars,  and  some  at  19  dollars  a  barrel  to  the 
amount  of  380  dollars,  how  many  barrels  did  he  sell  ? 

43.  A  drover  exchanged  42  horses  worth  72  dollars 
apiece,  for  cows  worth  36  dollars  apiece,  and  for  his  cows  he 
received  36  yoke  of  oxen,  which  he  sold  so  as  to  gain  144 
dollars,  how  much  did  he  get  for  each  yoke  of  oxen  ? 

44.  How  much  is  72  X  24  —  36  +  84  X  7  -5-  12  —  11  ? 

45.  Let  27  be  a  divisor,  and  567  a  dividend,  what  will  be 
the  quotient  ? 

46.  Suppose  25  is  a  quotient,  and  25  a  divisor  of  the  same 
dividend,  what  is  that  dividend  ? 

47.  Of  what  dividend  is  15  both  divisor  and  quotient  ? 


VI.    FRACTIONS. 

82.  Origin  of  Fractions,  and  Manner  of  Writing  them. 

1.    At  5  cents  a  quart  for  nuts,  how  many  quarts  can  you 
buy  for  38  c^nts  ? 

Since  1  quart  costs  5  cents,  you  can  buy  as  many  quarts  as 
there  are  times  5  cents  in  38  cents. 

5  is  contained  7  times  in  35,  which  being  subtracted  from 

38,  there  remain  3  units,  which  are 

5  )  38  (  7-|  quarts.         not  sufficient  to  contain  the  whole  of 

35  5,  but  if  %f  be  divided  into  5  equal 

—  parts,  each  part  is  exactly  a  unit ; 
3  therefore,  your  remainder  being  3 
3  units,  will  contain  exactly  3  of  the 

—  parts  of  5,  which  being  subtracted, 
nothing   remains.     In   the    quotient 

write  7  to  express  the  number  of  whole  5s ;  after  it  near  the 
top  write  a  small  3  to  express  the  number  ofparts^  and  under 
the  3,  separated  by  a  line,  write  a  small  5,  to  express  the 
size  of  these  parts,  which  it  will  do  by  showing  how  many 
such  parts  make  a  zmit,  or  a  whole,  of  which  these  are  parts. 
Hence,  as  38  contains  7  times  5  and  3  such  parts  of  a  5  that 
6  of  these  parts  would  make  a  whole  5,  you  can  buy  7  whole 


FRACTIONS.  65 

cuarts  and  3  such  parts  of  a  quart  that  5  of  them  would 
riake  a  whole  quart. 

Another  Explanation. 

35  cents  will  buy  7  quarts,  and  you  have  3  cents  remain- 
ing, which  are  not  sufficient  to  buy  a 
5)38  whole  quart:    but  if  a  whole  quart  be 

divided  into  5  equal  parts,  each  part  will 

I        7f  quarts.        be  worth  exactly  one  cent ;  and  as  you 
I  have  3  cents  remaining,  you  can  buy  3  of 

f  these  parts. 

Hence,  with  your  38  cents,  jou  can  buy  7  whole  quarts 
snd  3  such  parts  of  a  quart  that  5  of  them  would  make  a 
%irhole  quart. 

After  the  7  (whole  quarts)  near  the  top,  write  a  small  3 
to  express  the  number  of  parts,  and  under  the  3  separated 
by  a  line,  write  a  small  5  to  express  the  size  of  these  partSy 
"^v^hich  it  will  do  by  showing  how  many  such  parts  make  a 
unit,  or  whole  quart. 

2.  If  25  apples  be  given  to  7  boys,  what  would  be  the 
share  of  each  boy  ? 

Since  giving  one  apple  to  each  boy  takes  7  apples,  the 
share  of  each  boy  would  be  as  many  apples  as  there  are 
times  7  apples  in  25  apples. 

7  is  contained  3  times  in  21,  which  being  subtracted  from 

25,  there  remain  4  units,  which  are 

7 )  25  ( 3f-  apples.         not  sufficient  to  contain  the  ivhole  of 

21  7  ;  but  if  7  be  divided  into  7  equal 

I;       —  parts,  each   part  will   be   exactly  a 

W        4  unit;  therefore  the  remainder, being 

F        4  4  units,  will   contain   exactly  4  of 

►      —  these  parts  of  7,  which  being  sub- 

tracted, nothing  remains.  In  the 
quotient  write  3  to  express  the  number  of  ivhole  7s ;  after  it 
near  the  top  write  a  small  4  to  express  the  number  of  parts,  and 
under  the  4,  separated  by  a  line,  write  a  small  7  to  express  the 
size  of  these  parts,  which  it  will  do  by  showing  how  many 
such  parts  make  a  unit,  or  a  whole  7. 

Hence,  as  25  contains  3  times  7  and  4  such  parts  of  a  7 
that  7  of  them  would  make  a  whole  7,  liie  share  of  each  boy 
would  be  3  whole  apples  and  4  such  parts  of  an  apple  that  7 
of  them  would  make  a  whole  apple. 

6=^ 


I 


S6  ARITHMETIC. 

Another  explanation, 

21  apples  will  afford  3  to  each  boy ;  but  the  4  remaining 

apples  will  not  afford  the  boys  a  whole 

7  )  25  apple  apiece.    If,  however,  each  of  these 

4  apples  be  cut  into  7  equal  parts,  they 

3f  apples.         would  make  28  parts,  or  exactly  4  parts 
for  each  boy. 
Hence,  the  share  of  each  boy  would  be  3  whole  apples, 
and  4  such  parts  of  an  apple  that  7  of  them  would  make  a 
whole  apple. 

This  answer  may  be  expressed  as  before  directed. 

83.  Definition  of  Terms. 

A  fraction  is  the  expression  of  one,  or  more  of  the  eqtuil 
parts  of  a  unit, 

A  fraction  is  composed  of  two  numbers,  called  the  terms  of 
the  fraction. 

The  terms  of  a  fraction  are  written  one  below  the  other, 
separated  by  a  line. 

The  upper  term — called  the  numerator — shows  how 
many  parts  the  fraction  expresses. 

The  lower  term — called  the  denominator — shows  the 
size  of  the  parts  expressed  by  the  fraction,  by  showing  how 
many  such  parts  make  the  unit  of  which  the  fraction  ex- 
presses one  or  more  parts. 

84.  Manner  of  Reading  Fractions  and  Mixed  Numbers. 

Parts  take  different  names ^  according  to  their  size^  or  the 
number  of  them  that  it  takes  to  make  a  unit.  Thus,  the 
fractions  f ,  f ,  ^^,  y^,  &c.,  are  read,  two  thirds^  2  fifths,  two 
tenths,  2  hundredths,  &c. 

A  fraction  may  be  considered  and  read  in  four  different 
ways ;  for  instance,  |-  may  be  considered  f  of  1,  or  ^  of  3,  or 
3  divided  by  4,  or  3  such  parts  that  4  like  them  would  make 
a  unit. 

A  number  which  is  composed  of  both  an  integral  and 
fractional  number,  is  called  a  mixed  number.  The  answers  to 
the  above  problems  (82)  7f ,  3^,  are  mixed  numbers,  which 
are  read  thus,  seven  and  three  fifths,  three  and  four  sevenths. 

Integer  is  a  term  applied  to  a  number  which  expresses  only 
whole  units. 


FRACTIONS. 


07 


ml^iSm   Exercises  in  Originating  and  Writing  Fractions. 

11 

■     Solve  and  explain  the  following  prohleTus  on  the  left^  like 

the  firsts  and  those  on  the  rights  like  the  second  of  the  above 

vxamples  (82.) 

I.  If  1  lead  pencil  cost  3 
oents,  how  many  can  you  buy 

or  8  cents  ? 

3.  If  4  cents  buy  an  orange, 
.low  many  can  be  bought  for 
\IS  cents  ? 

5.  If  the  stage  fare  be  6 
cents  a  mile,  how  far  can  you 
ride  for  41  cents  ? 

7.  How  many  slates  at  8 
cents  apiece,  can  be  bought  for 
93  cents  ? 

9.  How  many  writing  books 
at  10  cents  apiece  can  be 
bought  for  125  cents  ? 

II.  How  many  shad  at  15 
cents  apiece,  can  be  bought 
for  218  cents  ? 

13.  At  22  cents  for  an 
inkstand,  how  many  may  be 
bought  for  93  cents  ? 

15.  At  29  dollars  a  head, 
how  many  cows  maybe  bought 
for  350  dollars  ? 

17.  How  many  acres  of  land 
at  37  dollars  an  acre,  will  5565 
dollars  buy  ? 

19.  If  320  rods  make  a 
mile,  how  many  miles  in 
46100  rods  ? 

21.  If  a  ship  sail  125  miles 
per  day,  ^how  long  would  it 
take  her  to  sail  round  the 
world,  it  being  24911  miles  ? 


2.  If  you  divide  11  lead 
pencils  among  3  boys,  how 
many  will  each  boy  have  ? 

4.  How  many  cents  does  1 
lemon  cost,  when  you  give  22 
cents  for  5  lemons  ? 

6.  How  much  does  a  man 
earn  a  week,  who  receives  65 
dollars  for  7  weeks  ? 

8.  If  9  men  do  a  job 
together,  and  receive  220  dol- 
lars, what  is  the  share  of  each? 

10.  What  does  a  single 
knife  cost,  at  295  cents  a 
dozen  ? 

12.  What  is  the  price  of  a 
barrel  of  flour,  when  18  barrels 
cost  150  dollars  ? 

14.  If  25  apple  trees  yield 
183  bushels  of  apples,  how 
much  does  each  tree  yield  ? 

16.  If  in  32  equal  loads  of 
potatoes  729  bushels  were  car^ 
ried  to  market,  how  many 
bushels  in  each  load  ? 

15.  If  1760  yards  make  320 
rods,  how  many  yards  make 
Irod? 

20.  If  1749  feet  make  106 
rods,  how  many  feet  in  one 
rod? 

22.  If  a  ship  sail  132  miles 
a  day,  in  how  many  days  will 
she  sail  Jrom  Boston  to  Liver- 
pool, it  being  3000  miles  ? 


I 


ARITHMETIC. 


24.  What  would  be  the  cost 
of  1  hogshead  of  molasses,  if 
75  hogsheads  cost  2200  dol- 
lars? 

26.  How  many  bushels  of 
wheat  does  a  farmer  raise 
on  an  acre,  who  raises  2400 
bushels  on  99  acres  ? 

28.  If  a  man  receive  730 
dollars  a  year,  how  much  is 
that  a  week  ? 

30.  How  many  miles  per 
hour  does  an  engine  move, 
which  goes  2600  miles  in  a 
week? 

32.  Divide  4657  into  25 
equal  parts. 

34.  Divide  100000000  by 
12478. 


23.  If  63  gallons  of  water 
in  one  hour  run  into  a  cistern 
containing 432  gallons,  in  what 
time  will  it  be  filled  ? 

25.  How  many  boxes  would 
be  required  to  contain  32844 
oranges,  if  each  box  contain 
exactly  100  oranges  ? 

27.  How  many  days,  at  175 
cents  a  day,  must  a  man  work 
to  earn  4500  cents  ? 

29.  At  365  days  a  year, 
how  many  years  old  is  a  boy 
who  has  lived  3999  days  ? 

31.  How  many  times  199 
in  2569? 

33.    Divide  2864  by  14. 

8©.   Observation. 

Observe,  (S^j)  that  when  a  number  is  to  be  divided  which 
is  smaller  than  the  divisor,  the  quotient  will  be  a  fraction,  of 
which  the  dividend  will  he  the  numerator  and  the  divisor  will 
be  the  denominator. 

Hence,  division  may  be  expressed  in  a  fractional  form, 
whether  the  dividend  be  larger  or  smaller  than  the  divisor, 
and  the  value  of  the  expression  will  be  the  true  quotient. 

87.    Model  of  a  Recitation. 

1.  If  a  pie  be  cut  into  8  equal  parts,  what  fractions  would 
express  one,  three,  five,  and  eight  of  the  parts  ? 

When  8  equal  parts  make  a  unit,  any  number  of  these 
parts  are  so  many  eighths ;  (84)  therefore,  one  part  is  -J- 
(one  eighth,)  three  parts  are  f  (three  eighths,)  5  parts  are 
|-  (five  eighths,)  and  eight  parts  are  f  (eight  eighths,)  or  the 
whole  pie. 

2.  What  fractions  of  a  foot  will  express  5,  7,  and  11 
inches  ? 

When  a  unit  is  divided  into  12  equal  parts,  any  number 
of  the  parts  are  so  many  twelfths,  (84,)  therefore,  5  inches 


FRACTIONS.    .  69 

aie  1^  of  a  foot,  7  inches  are  xV  of  a  foot,  and  11  inches 
ai  e  \^  of  a  foot, 

3.    What  parts  of  15  are  8,  14,  and  19  ? 

Since  it  takes  15  units  to  make  the  whole  of  15,  any  num- 
b(;r  of  units  are  so  many  fifteenths  of  15 ;  therefore,  8  is  -^ 
01'  15, 14  is  if  of  15,  and  19  is  |f  of  15. 

88.   Exercises  in  Expressing  Division. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  If  a  pie  be  cut  into  6  equal  pieces,  what  fractions  will 
express  one,  two,  and  five  of  the  pieces  ? 

2.  Two  boys  divided  an  orange  equally  between  them- 
selves, what  fraction  will  express  each  one's  part  ?    * 

3.  If  an  acre  of  land  be  divided  into  4  equal  house-lots, 
what  fractions  would  express  one,  three,  and  four  of  the  lots  ? 

4.  If  a  piece  of  cloth  be  sufficient  for  7  coats,  what  parts 
01  the  piece  of  cloth  would  be  sufficient  for  1, 3, 5,  and  6  coats  ? 

5.  If  you  divide  a  barrel  of  flour  equally  among  9  men, 
what  part  of  a  barrel  would  each  receive  ? 

6.  If  IS  dollars  be  paid  for  a  ton  of  hay,  what  parts  of  a 
ton  may  be  bought  for  5,  7,  11,  and  17  dollars  ? 

7.  At  27  dollars  a  hogshead  for  molasses,  what  parts  of  a 
hogshead  may  be  bought  for  10,  14,  19,  and  25  dollars? 

8.  At  one  hundred  dollars  a  share  in  a  bank,  what  parts 
of  a  share  may  be  bought  for  16,  29,  67,  89,  and  93  dollars. 

9.  At  75  cents  a  bushel  for  corn,  what  parts  of  a  bushel 
may  be  bought  for  12,  24,  36,  and  58  cents  ? 

10.  What  fractions  of  a  dollar  will  express  7,  23,  37,  47, 
67,  and  97  cents  ? 

11.  What  fractions  of  June  will  express  11,  17,  and  29 
days  ? 

12.  What  parts  of  July  are  16,  21,  and  27  days  ? 

13.  What  parts  of  an  hour  are  13,  43,  and  59  minutes  ? 

14.  What  parts  of  a  day  are  1,  7,  19,  and  23  hours  ? 

15.  11,  21,  87,  123,  and  219  rods  are  what  parts  of  a 
mile  ? 

16.  What  part  of  5  dollars  are  3  dollars  ? 

17.  What  parts  of  25  cents  are  3,  7,  14,  and  21  cents  ? 

18.  What  parts  of  63  gallons  are  16,  31,  and  44  gal- 
lons ? 


70  ARITHMETIC. 

19.  What  parts  of  365  days  are  31, 60,  124,  243,  and  316 
days  ? 

20.  15  weeks  are  what  part  of  52  weeks  ? 

21.  What  fractions  of  a  bushel  will  express  11,  15,25 
and  32  quarts  ? 

22.  What  part  of  8  is  5? 

23.  What  parts  of  11  are  2,  9,  12,  14,  and  21  ? 

24.  What  parts  of  33  are  5,  7,  16,  25,  and  32  ? 
•  25.    How  many  times  is  15  contained  in  34  ? 

26.  How  many  times,  or,  (more  properly,)  what  part  of  a 
time,  is  15  contained  in  8  ? 

27.  What  part  of  a  time  is  24  contained  in  7  ? 

28.  What  part  of  16  is  contained  in  11  ? 

29.  What  part  of  12  does  5  contain  ? 

30.  Divide  21  by  25 ;  what  is  the  quotient  ? 

31.  If  17  be  a  dividend,  and  25  the  divisor,  what  must  be 
the  quotient  ? 

32.  If  4  apples  be  divided  among  5  boys,  what  part  of  an 
apple  is  each  boy's  share  ? 

33.  If  3  men  divide  a  barrel  of  apples  equally  among 
themselves,  what  fractions  will  express  the  shares  of  1,  2, 
and  3  men  ? 

34.  If  15  bushels  of  potatoes  cost  7  dollars,  what  part  of  a 
dollar  would  1  bushel  cost  ? 

35.  If  2  bushels  of  wheat  sow  3  acres,  what  part  of  a 
bushel  would  sow  1  acre  ? 

36.  If  a  cord  of  wood  last  7  weeks,  what  part  of  a  cord 
would  last  1  week  ? 

37.  Divide  16  by  17  ;  what  is  the  quotient  ? 

38.  If  2  be  a  dividend,  and  21  the  divisor,  what  must  be 
the  quotient? 

39.  Divide  17  by  123. 

40.  Divide  84  by  1725. 

41.  Divide  1728  by  1837. 

42.  Express  the  division  of  37  by  25. 

43.  Express  the  division  of  25  by  36. 

44.  Express  the  division  of  81  by  75. 

45.  Express  the  division  of  16  by  9. 

46.  Divide  7  by  11. 

89.   Model  of  a  Recitation. 

1.  If  a  man  receive  125  dollars  for  J  of  his  annual  salary, 
what  is  his  salary  ? 


FRACTIONS.  71 

Since  J  of  anything  make  the  whole  of  that  thing,  (83j) 
if  \  of  his  salary  is  125  dollars,  |,  or  the  whole  of  his  salary, 
will  be  4  times  125  dollars,  equal  to  500  dollars,  which  is  the 
answer  required. 

2.    18  is  :J  of  what  number  ? 

Since  f  of  any  number  make  the  whole  of  that  number, 
if  one  eighth  oi  some  number  is  18,  the  whole  of  that  num- 
ber will  be  8  times  18,  equal  to  144,  which  is  the  answer 
re  quired. 

9<l>,   Exercises  in  Finding  the  Whole   of  a  Quantity 

FROM   a    single     PaRT    OF    IT. 

In  like  manner^  solve  and  explain  the  following  problems. 

1.  If  ^  of  a  bushel  of  corn  cost  42  cents,  what  is  that  a 
bushel  ? 

2.  42  is  j-  of  what  number  ? 

3.  If  -J  of  an  acre  produce  23  bushels  of  corn,  how  many 
bushels  would  1  acre  of  land  produce  ? 

4  23  is  ^  oi  what  number  ? 

5.  If  ^  of  the  annual  rent  of  a  house  be  75  dollars,  how 
much  is  that  for  a  year  ? 

6.  75  is  ^  of  what  number  ? 

7.  25  is  I  of  what  number  ? 

8.  33  is  ^  of  what  number  ? 

9.  16  is  ^  of  what  number  ? 

10.  If  ^  of  a  mile  is  40  rods,  how  many  rods  in  a  mile  ? 

11.  If  J  of  a-  hogshead  be  7  gallons,  how  many  gallons  in 
that  hogshead  ? 

12.  If  ^^  of  an  acre  is  4  square  rods,  how  many  square 
rods  in  an  acre  ? 

13.  If  60  minutes  be  ^V  ^^  3,  day,  how  many  minutes  in  a 
day  ? 

14.  If  1  day  be  -r^-^  of  a  year,  how  many  days  in  a  year  ? 

15.  At  35  dollars  for  working  -^  of  a  year,  how  much  is 
that  for  a  year  ? 

16.  If  25  cents  make  J-  of  a  dollar,  how  many  cents  in  a 
dollar  ? 

17.  62  is  ^  of  what  number  ? 

18.  18  is  5^  of  what  number  ? 

19.  John,  being  12  years  old,  was  only  ^  as  old  as  his 
grandfather.     How  old  was  John's  grandfather  ? 


72  arithmetic. 

91*   Model  of  a  Recitation. 

1.  John  having  100  cents,  paid  away  ^  of  them  for  a  pen- 
knife.    How  many  cents  did  his  penknife  cost  ? 

Since  it  takes  four  ^s  of  any  thing,  or  numher,  to  make 
the  whole  of  it,  (SO,)  if  100  be  divided  by  4,  the  quotient 
will  be  ^  of  100  cents,  equal  to  25  cents,  which  is  the  answer 
required. 

2.  What  is  i  of  144  ? 

Since  it  takes  |  of  144  to  make  the  whole  of  it,  divide  144 
by  8,  and  the  quotient  will  be  ^  of  144,  equal  to  18,  which  is 
the  answer  required. 

93«   Observation. 

Observe,  that^  the  dividerid  being  the  proddci  of  the  divisor 
and  quotient^  C^,)  the  divisor  shows  how  many  equal  parts ^ 
such  as  the  quotient,  (6S5)  will  make  the  dividend. 

Therefore,  to  ascertain  any  single  part  of  a  number, 
divide  it  by  the  numher  which  shows  how  many  such  parts 
make  the  integer,  or  given  number, 

93«   Exercises  in   Finding   a   single    Part  of  a  Quan- 
tity from  the  Whole  of  it. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  If  ^  of  100  cents  be  paid  for  a  penknife,  how  many 
cents  would  the  penknife  cost  ? 

2.  How  many  cents  in  ^  of  a  dollar  ? 

3.  How  many  cents  in  |  of  a  dollar  ? — in  ^  of  a  dollar  ? — 
in  |.  of  a  dollar  ? — in  ^  of  a  dollar  ? — in  -^^  of  a  dollar  ? — 
in  ^  of  a  dollar  ? 

4.  If  a  ton  of  hay  cost  21  dollars,  what  would  if  of  a  ton 
cost  ? 

5.  What  is  \  of  63  ?— of  72  ?— of  81  ?— of  90  ?— of  99  ? 
—  of  108? 

6.  What  is  -^  of  each  of  the  following  numbers  :  11,  22, 
33,  44,  ^^,  99,  and  132  ? 

7.  If  a  man,  owning  279  acres  of  land,  sell  ^  of  it ;  how 
many  acres  would  he  sell  ? 

8.  If  160  square  rods  make  an  acre,  how  many  rods  in  \ 
of  an  acre  ? 

9.  If  320  rods  make  a  mile  in  distance,  how  many  rods 
in  i  of  a  mile. 


FRACTIONS.  73 

10.  A  furlong  being  ^  of  a  mile,  how  many  rods  in  a  fur- 
long ? 

11.  In  a  day  there  are  1440  minutes.    How  many  minutes 

12.  In 'a  pound  there  are  960  farthings.  How  many  far- 
things in  a  shilling,  which  is  ^V  of  a  pound  ? 

13.  If  a  slaughtered  ox  weigh  896  pounds,  what  would  be 
th  e  weight  of  each  quarter,  the  quarters  being  equal  ? 

14.  A  man  hired  a  farm  "  at  the  halves,"  and  raised  624 
bushels  of  potatoes,  150  bushels  of  rye,  64  bushels  of  wheat, 
7o  bushels  of  oats,  12  bushels  of  white  beans,  50  bushels  of 
turnips,  25  bush,  of  corn,  45  bush,  of  winter  apples,  and  40 
bushels  of  sauce  apples.  How  many  bushels  in  his  share  of 
this  produce  ? 

15.  If  you  could  buy  480  apples  for  a  dollar,  how  many 
could  you  buy  up  for  ^  of  a  dollar  ? — for  J-  of  a  dollar  ? — for 
4^  of  a  dollar  ? — for  -j^  of  a  dollar  ?  . 

16.  If  a  man's  salary  be  800  dollars  a  year,  how  much  is 
that  for  I  of  a  year  ? — for  J  of  a  year  ? — for  ^^  <^^  ^  7®^^  • 

17.  If  32  quarts  of  nuts  be  divided  equally  among  4  boys, 
what  part,  and  how  much  of  them,  is  each  boy's  share  ? 

18.  Divide  64  by  16 ;  what  part  of  64  is  the  quotient  ? 

19.  If  you  divide  any  number  by  4,  what  part  of  that 
number  will  be  the  quotient? 

20.  What  is  ^  of  1000  dollars  ? 

d4.   Exercises  in  the  Different  Modes  of  Considering 

AND  Reading  Fractions. 

J  of  I  is  J,  and  |  of  3  is  three  times  as  much,  or  |  of  1 ; 
i  of  5  is  J  of  1 ;  1  of  13  is  Y"  of  1 ;  -J  of  3  is  f  of  1 ;  |  of 
5  is  -f  of  1. 

1.  How  many  ninths  of  1  is  ^  of  7  ? 

2.  J  of  10  is  how  many  thirds  of  1  ? 

3.  i  of  11  is  what  part  of  1  ? 

4.  W  hat  part  of  1  is  ^^  of  25  ? 

5.  Read  the  following  fractions  in  the  four  different  modes 
described  (84). 

h  /t>  a.  h  ih  M.  V-»  i^F^  T%^  ^^  ih  ih  M'  V»  »»  h 
ilh  h  h  h  A»  A»  V- 

6.  Which  of  these  fractions  expresses  the  greatest  number 
of  parts  ? 

7 


74  ARITHMETIC. 

7.  Which  expresses  the  largest  parts  ? 

8.  Which  expresses  the  smallest  parts  ? 

9.  Which  expresses  the  smallest  number  of  parts  ? 

10.  Which  express  the  same  number  of  parts  ? 

11.  Which  express  parts  of  the  same  size  ? 

12.  Which  express  just  parts  enough  to  make  a  unit? 

13.  Which  express  parts  enough  to  make  more  than  one 
unit? 

14.  Considering  both  the  number  and  size  of  the  parts, 
which  is  the  largest  fraction  ? 

15.  Which  is  the  smallest  fraction  ? 

16.  Why,  of  two  fractions  having  equal  denominators,  is 
that  greatest  which  has  the  greatest  numerator  ? 

17.  Why,  of  two  fractions  having  equal  numerators,  is 
that  greatest  which  has  the  smallest  denominator  ? 

18.  What  effect  is  produced  upon  the  value  of  a  fraction 
by  diminishing  its  numerator  ?  , 

19.  What  effect  is  produced  upon  the  value  of  a  fraction 
by  increasing  its  denominator  ? 

95#    Expression,   Definition,  and   Reduction  of  an  Im- 
proper Fraction. 

As  there  is  no  limit  to  the  number  of  parts  that  may  be 
expressed  by  a  fraction,  (885)  it  is  often  convenient  to  ex- 
press in  one  fraction,  more  parts  than  there  are  of  that  size, 
in  one  tmit. 

But  a  fraction  whose  value  is  equal  to^  or  greater  than  its 
unit  J  is  called  an  improper  fraction ;  and  a  fraction  whose 
value  is  less  than  its  unit,  is  called  a  proper  fraction. 

The  value  of  a  fraction  being  the  quotient  resulting  from 
the  division  of  its  numerator  by  its  denominator,  (865)  an 
improper  fraction  may  be  reduced  (208)  to  its  equal  inte- 
gral, or  mixed  number,  by  performing  the  division,  which  is 
only  expressed  by  the  fraction-. 

96.   Model  of  a  Recitation. 

1.  A  toll-gatherer  took  in  one  week  ^-f-^  of  a  dollar,  (four- 
pence-half-pennies  ; )  how  many  dollars  would  they  make  ? 

Since  there  were  165  such  parts  of  a  dollar,  that  every  16 
of  them  would  make  a  dollar,  (895)  they  would  make  as 
many  dollars  as  there  are  times  16  in  165.     Thus : 

-y^=  10^  dollars,  which  is  the  answer  required. 


fractions.  75 

9?.  Exercises  in  the  Reduction  of  Improper  Fractions. 

In  like  manner^  solve  and  explain  the  following  problems, 

1.  At  a  certain  contribution,  ^^^  of  a  dollar  (ninepences) 
were  taken  ;  how  many  dollars  were  taken  ? 

2.  A  merchant  sold  calico  for  ^  of  a  dollar  a  yard,  till  he 
re  ceived  -^^  of  a  dollar ;  how  many  dollars  did  he  receive  ? 

3.  At  a  large  party,  ^^-  of  a  pie  were  eaten,  how  many 
whole  pies  were  eaten  ? 

4.  In  ^-f^-  of  a  bushel  how  many  bushels  ? 

5.  In  -^jV-  of  a  pound  how  many  pounds  ? 

6.  In  4¥-  ^^  ^  shilling  how  many  shillings  ? 

7.  In  -^Z/-  of  a  guinea  how  many  guineas  ? 

8.  In  -^Jf  ^  of  a  day  how  many  days  ? 

9.  In  ^^%^  of  an  hour  how  many  hours  ? 

10.  In  ^f  f  ^^  of  a  year  how  many  years  ? 

11.  Reduce  -VV"  ^^  units. 

12.  Reduce  -ViV-  to  an  integral  number. 

13.  Reduce  \^-  to  a  mixed  number. 

14.  Reduce  ^^■^-  to  a  mixed  number. 

15.  Reduce  -yf-  to  an  integral,  or  mixed  number. 

16.  Change  -*f f ^   to  an  integral,  or  mixed  number. 

17.  Change  |f|  to  an  integral,  or  mixed  number. 

18.  Reduce  -fi^  to  an  integral,  or  mixed  number. 

19.  How  many  units   in  Mj^-^? 

20.  What  mixed  number  is  equal  to  -If  f  ? 

21.  What  is  the  value  of  |^ff  in  a  mixed  number? 

08.   Model  of  a  Recitation. 

1.    Reduce  5^-^  to  an  improper  fraction,  that  is,  to  six- 
teenths. 

Since  there  are  16  sixteenths  in  one  unit,  there  will  be  16 
times  as  many  sixteenths  as  units  in  any  number. 

16  times  5  are  SO  sixteenths y  and  the  other  3  sixteenths 

-^^ 3  3  are   ff,  which   is   the   answer   re- 

^      ^^'  quired. 

S>9.   Exercises  in  Reducing  Integral  and  Mixed  Num- 
bers TO  Fractions.' 

In  like  manner,  solve  and  explain  the  following  probUTns, 

1.  Reduce  7  to  sixteenths. 

2.  Reduce  25f  to  an  improper  fraction. 


76  ARITHBIETIC. 

3.  Change  ]2f  to  an  improper  fraction. 

4.  What  fraction  is  equal  to  3^^  ? 

5.  What  is  10|-  equal  to  in  a  fractional  form  ? 

6.  13|  are  how  many  ninths  ? 

7.  How  many  eighths  of  one  dollar  are  9  zi?AoZc  dollars  ? 

8.  How  many  Js  of  a  yard  are  32  yards  ? 

9.  1 5^1  days  are  how  many  ^^s  of  a  day  ? 

10.  82^^  pounds  are  how  many  -^^  of  a  pound  ? 

11.  17|^  hours  are  equal  to  how  many  -^s  of  an  hour? 

12.  6^§  hogsheads  are  equal  to  how  many  ^s  of  a 
hogshead  ? 

13.  How  many  g-^-^s  of  a  year  are  equal  to  10  years  ? 

14.  Keduce  437^y  to  an  improper  fraction. 

15.  Reduce  lO^f-^-  to  an  improper  fraction. 

16.  Reduce  25^^^^  to  an  improper  fraction. 

17.  What  fraction  is  equal  to  50-^  ? 

18.  Change  20  to  sevenths. 

19.  Reduce  36  to  twelfths. 

20.  Reduce  15  to  fifths ;  also  to  sixths. 

21.  Change  4  to  halves,  to  thirds,  to  fourths,  to  fifths,  and 
to  sixths. 

22.  Reduce  16  to  halves,  to  thirds,  to  fourths,  and  to  fifths. 

23.  Reduce  1  to  halves,  to  fifteenths,  and  to  seventy-fifths. 

24.  Reduce  1  to  halves,  thirds,  fourths,  fifths,  sixtfis,  and 
to  sevenths. 

100  •    Model  of  a  Recitation. 

1.  A  man  bought  25  yards  of  calico,  at  -3%  of  a  dollar  (3 
fourpence-half-pennies)  a  yard ;  how  many  dollars  did  his 
calico  cost  ? 

Since  1  yard  cost  ^^  of  a  dollar,  25  yards  would  cost  25 

^^=X5  =  4i ,  dollars.  *T''  f  ^f  ^^2/ f  ?^f  ^^>^^ 

^^  ^^         ^^  of  a  dollar,  which   are 

•J-f  of  a  dollar ;    equal    to  4^^  dollars,  ( 95j )  the  answer 

required. 

101  •    Observation. 

Observe,  (lOOj)  that  in  multiplying  the  numerator 
ONLY  by  25,  retaining  the  same  denominator ,  you  multiply 
the  fraction;  for  thus,  you  produce  25  times  as  many  parts 
(  83 )  of  the  same  size. 


FRACTIONS.  77 


103,   Exercises  in   Multiplying  a  Fraction  by  an  In- 
tegral Number. 

In  like  TTianner^  solve  and  explain  the  following  problems, 

1.  How  many  dollars  will  25  penknives  come  to,  at  f  of  a 
dollar  apiece? 

2.  How  many  dollars  would  pay  a  man  to  work  5  days,  at 
^  of  a  dollar  per  day  ? 

3.  How  many  dollars  should  Mr.  Farmer  receive  for  12 
bushels  of  corn,  at  f  of  a  dollar  a  bushel  ? 

4.  At  y*2-  of  a  dollar  a  pound  for  beef,  how  much  would  1 1 
pounds  cost? 

5.  If  a  family  consume  |  of  a  barrel  of  flour  in  a  week, 
how  much  flour  would  last  them  a  year? 

6.  If  it  take  |  of  a  bushel  of  rye  to  sow  an  acre,  15  acres 
would  require  how  many  bushels  ? 

7.  If  a  horse  eat  ^  of  a  bushel  of  oats  in  a  day,  how  much 
would  keep  him  through  December  ? 

8.  If  1  bushel  of  apples  cost  ^  of  a  dollar,  what  would  be 
the  value  of  a  load  containing  33  bushels  ? 

9.  At  ^  of  a  dollar  a  day  for  board,  what  would  be  the 
cost  of  board  for  365  days  ? 

10.  How  far  can  I  ride  in  1  hour  at  the  rate  of  -^  of  a 
mile  per  minute  ? 

11.  How  much  is  5  times  -J-^  ? 

12.  Multiply  /^  by  13. 

13.  Multiply  tJ^  by  43. 

14.  Multiply  VbV  by  36. 

15.  Multiply  f  ^  by  3. 

16.  How  much  is  15  times  ^|-  ? 

17.  Multiply  ^^^  by  366. 

18.  How  much  is  3  times  x^Jg-  ? 

103.   Model  of  a  Recitation. 

1.    At  32f  dollars  apiece,  what  would  7  cows  cost? 
095  Since  1  cow  cost  32|^  dollars,  7  cows 

17^  would  cost  7  times  32f . 

Seven  times  f  are  ^,  equal  to  f  which 


2283  dolls  write,  and  4  units,  which  add  with  the 

V  ^  '  units,  &c.  (27) 

2.    How  much  is  83  times  16f  feet  ? 
It  will  be  most  convenient,  in  this  example,  to  multiply  the 
7# 


78  ARITHMETIC. 

integral  and  fractional  parts  separately,  and  add  the  products 
together.     Thus : 
16§ 

48  ?^-a^  =  J4&  =  55^  feet. 

128 

1383^  feet. 

104.   Exercises  in  Multiplying  a  Mixed   Number  by 
AN  Integral  Number. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  If  15  yards  are  sufficient  for  one  coat,  how  many  yards 
will  be  sufficient  for  10  coats  ? 

2.  How  many  feet  in  25  rods,  there  being  16^  feet  in  1  rod  ? 

3.  How  many  yards  in  40  rods,  there  being  5^  yards  in 
Irod? 

4.  How  many  cents  in  6  shillings,  there  being  16f  cents 
in  1  shilling? 

5.  How  old  is  John,  if  he  is  3  times  as  old  as  Charles, 
and  Charles  is  3^V5  years  old  ? 

6.  What  would  be  the  cost  of  15  barrels  of  flour,  at  6| 
dollars  per  barrel  ? 

7.  If  31^  gallons  make  a  barrel,  how  many  gallons  in  50 
barrels  ? 

8.  What  is  the  price  of  a  dozen   bibles   at   2f  dollars 
apiece  ?  ^ 

9.  What  is  the  cost  of  10  dozen  pairs  of  shoes  at  1| 
dollars  a  pair  ? 

10.  What  would  7  tons  of  Lehigh  coal  cost  at  9^  dollars 
a  ton? 

11.  What  would  17  grind-stones  come  to  at  3-i^  dollars 
apiece  ? 

12.  Multiply  6|-^  by  35. 

13.  How  much  is  100  times  2^  ? 

14.  What  is  the  product  of  lf|  multiplied  by  5  ? 

15.  How  much  is  16f  X  10  ? 

16.  Multiply  1728^^^^  by  7. 

10«l«    Model  of  a  Recitation. 

1.    At  -^^  of  a  dollar  (3  fourpence-half-pennies)  apiece, 
what  would  be  the  postage  of  4  letters  ? 


FRACTIONS.  7& 

Since  the  postage  of  1  letter  is  -^^  of  a  dollar,  the  postage 
oi  4  letters  would  be  4  times  as  much. 

This  product  can  be  ascertained,  either  by  multiplying  the 
niTnerator by  4, retaining  the  same  denominator  ;  (lOlj)  or, 
far  better,  by  dividing  the  denominator  by  4,  retaining  the 
seme  numerator. 

For,  by  the  former  process,  you  make  the  number  of  parts 
4  times  as  large,  the  parts  retaining  the  same  size  ;  and,  by 
the  latter  process,  you  make  the  size  of  the  parts  4  times  as 
krge,  retaining  the  same  number  of  parts. 

It  is  evident  that,  by  the  latter  process,  the  parts  are  made 
4  times  as  large,  from  the.  fact  that,  it  will  take  only  J  as 
71  any  of  them  to  make  the  unit  as  before, 

-ff  of  a  dollar  are  12  fourpence-half-pennies,  and  J  of  a 
dollar    are  also   12   fourpence- 

3  XA_  1 2  of  a  dollar  half-pennies ;    for  ^  of  a  dollar 

^^  *^  '  is  equal   to   4    fourpence-half- 

^3^^  =  I  of  a  dollar.  f  "^,^^'  ^""^  *  "^  ""  dollar  will 

^^-*       ^  be  3  times  as  majiy,  or  12  four- 

pence-half-pennies. 
The  two  processes  giving  the  same  result,  the  latter  is  to 
be  adopte^  in  all  cases  when  the  multiplier  is  a  factor  (25) 
of  the  denominator ;  because  it  will  give  the  result  in  lower 
terms,  |-  being  in  lower  terms,  and,  consequently,  a  more 
simple  fraction  than  its  equal  -^f. 

2.  At  f  of  a  dollar  a  bushel,  what  would  be  the  price  of  8 
bushels  of  potatoes  ? 

Since  the  price  of  1  bushel  is  f  of  a  dollar,  the  price  of  8 

bushels  would  be   8   times   as 

f q:^  =  f  =  3  dollars.  much,  which   is  3  dollars,  the 

answer  required. 
For,  by  dividing  the  denominator  by  8,  the  parts  become 
8  times  as  large,  and  such  that  each  one  of  them  makes  a  unit. 

106.    Observation. 

Observe,  (lOS^)  that  by  whatever  number  the  denomina- 
tor IS  DIVIDED,  retaining  the  same  numerator,  the  fraction 
IS  THUS  multiplied  BY  THAT  NUMBER ;  ^r  the  denominator 
shoioing  the  number  of  parts  that  make  a  unit,  their  size  is 

INCREASED    IN    THE  'SAME    RATIO  .THAT     THE     DENOMINATOR     IS 
DIMINISHED. 

Observe,  also,  that  if  a  fraction  be  multiplied  by  its  denomi- 
viator,  the  product  will  be  the  numerator. 


80  ARITHMETIG. 


107*    Exercises  in  Multiplying  a  Fraction  by  dividing 
ITS  Denominator. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  If  1  yard  of  calico  cost  |^  of  a  dollar,  what  would  be  the 
cost  of  2  yards  ?  What  would  be  the  cost  of  4  yards  ? 

2.  At  J  of  a  dollar  a  pound,  what  will  2  pounds  of  butter 
cost  ?     What  will  3  pounds  cost  ?     What  will  6  pounds  cost  ? 

3.  At  |-  of  a  dollar  apiece,  what  would  be  the  postage  of  2 
letters  ?  of  4  letters  ?  of  8  letters  ? 

4.  If  1  ninepence  is  J  of  a  dollar,  what  part  of  a  dollar  is 
2  ninepences  ?  is  4  ninepences  ?  is  8  ninepences  ? 

5.  If  1  fourpence-half-penny  is  -^^  of  a  dollar,  what  part 
of  a  dollar  is  2  fourpence-half-pennies  ?  is  4  fourpence-half- 
pennies  ?  is  8  fourpence-half-pennies  ?  is  16  fourpence-half- 
pennies  ? 

6.  If  a  sixpence  is  -^  of  a  dollar,  what  part  of  a  dollar  is 
2  sixpences  ?  is  3  sixpences  ?  is  4  sixpences  ?  is  6  sixpences  ? 
is  12  sixpences*? 

7.  At  |-  of  a  yard  apiece  for  vests,  how  much  satin  would 
be  necessary  for  3  vests  ?  for  9  vests  ? 

8.  At  f  §  of  a  mile  a  minute,  how  far  would  a  train  of  cars 
run  in  2  minutes  ?  in  3  minutes  ?  in  4  minutes  ?  in  5 
minutes  ?  in  6  minutes  ?  in  10  minutes  ?  in  12  minutes  ?  in 
15  minutes  ?  in  20  minutes  ?  in  1  hour  ? 

9.  If  it  take  l^f  yards  of  broad  cloth  to  make  a  coat,  how 
much  would  it  take  for  3  coats  ?  for  6  coats  ?  for  8  coats  ? 
for  12  coats  ?  for  24  coats  ? 

10.  Multiply  ^  by  5. 

11.  Multiply  ft  by  7. 

12.  Muhiply  -J^^  by  25. 

13.  Multiply  ^5%  by  100. 

14.  Multiply  II  by  16. 

15.  Multiply  7|  by  4. 

16.  Multiply  4^11  by  365. 

17.  How  much  is  20  times  9^  ? 

18.  How  much  is  327  times  10^  ? 

19.  J/-  is  -1  of  what  number  ? 

20.  5f  is  ^  of  what  number  ? 

21.  125^3^  is  yV  o^  what  number  ? 

22.  ^-  is  I  of  what  number  ? 

23.  Multiply  -^-^  by  5,  and  that  product  by  3 


it 


FRACTIONS.  81 


?H24.   Multiply  -^  by  5,  and  that  product  by  5. 
25.    Multiply  j\  by  3,  and  that  product  by  5. 

1.08.   Model  of  a  Recitation. 

1.    Multiply  5^  by  36. 

Since  36  is  not  a  factor  of  the  denominator,  but  9,  one  of 

the  factors  of  3.6,  is  also  a  factor  of  the 

^^^4  =  JJ§.=  3|.    denominator,  multiply  first  by  9,  (^O,) 

by  making  the  parts  9  times  as  large, 

(IO65)  and  then  multiply  that  product  by  4,  the  other  factor 

of  36,  by  making  4  times  as  many  parts,  (IOI5)  which  will 

give  4  times  9  times,  or  36  times  ^,  equal  to  -^5^,  equal  to 

2^,  which  is  the  product  required. 

1.09*    Exercises   in    Multiplying  a  Fraction   by   the 
Factors  of  the  Multiplier. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  Multiply  ^  by  18. 

2.  Multiply  4^  by  35. 

3.  Multiply  II  by  48. 

4.  How  much  is  24  times  ^?  * 

5.  How  much  is  50  times  f  f  ? 

6.  How  much  is  81  times  ^  ? 


110«   Model  of  a  Recitation. 

1.  If  3  yards  of  calico  cost  ^^  of  a  dollar,  (9  fourpence- 
half-pennies,)  what  would  be  the  price  of  1  yard  ? 

Since  1  yard  is  -J  of  3  yards,  the  price  of  1  yard  must  be 

■^  of  the  price  of  3  yards. 
^^=-2.=^  of  a  dollar.  Therefore,  as  the  price  of  3 

yards  is  -^^  of  a  dollar,  the 
price  of  1  yard  will  be  ^  (92)  as  Tnany  sixteenths  of  a  dollar, 
equal  to  y^^,  the  answer  required. 

111.  Observation. 

Observe  that,  m  dividing  the  numerator  only  by  3,  retain- 
ing the  same  denomiTudor,  you  divide  the  fraction  ;  for  thus 
you  obtain  \  as  many  parts  (83)  of  the  same  size. 

112.  Model  of  a  Recitation. 

1.  At  6  dollars  a  barrel,  how  many  barrels  of  flour  may 
be  bought  for  45^^  dollars  ? 


83  ARITHMETIC. 

Since  1  barrel  costs  6  dollars,  you  may  buy  ^  (09)  as 

many  barrels  as  you  have  dollars  ;  -J 

6  )  45i\  of  42  is  7.     Reduce    the   remaining 

3  to    elevenths    making    33,    these 

7^  barrels.         and  the  other  3  elevenths  are  ff ,  ^ 
of  which  are  ^y  of  a  barrel,  which 
written  with  the  7  barrels  make  7-j-\  barrels,    the    answer 
required. 

1 13.  Exercises  in  Dividing  a  Fraction  by  an  Integral 
Number. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  If  2  bushels  of  potatoes  cost  f  of  a  dollar,  what  would 
that  be  a  bushel  ? 

2.  If  a  cow  consume  |-  of  a  bushel  of  meal  in  3  days,  how 
much  would  that  be  per  day  ? 

3.  At  1%  of  a  dollar  for  4  pounds  of  beef,  what  would  be 
the  cost  of  1  pound  ? 

4.  If  4  horses  consume  ^f  of  a  ton  of  hay  in  a  month, 
how  much  would  that  be  for  1  horse  ? 

5.  At  14-  of  a  dollar  for  7  pounds  of  coffee,  what  would  be 
the  cost  of  1  pound  ? 

6.  If  2  yards  of  cloth  cost  8|  dollars,  wha<  would  1  yard 
cost  at  that  rate  ? 

7.  What  would  be  the  cost  of  1  bushel  of  wheat,  if  4 
bushels  cost  32|-  shillings  ? 

8.  If  I  give  23|  bushels  of  wheat  for  3  sheep,  how  much 
would  that  be  apiece  ? 

9.  If  I  give  59^  bushels  of  corn  for  7  calves,  how  many 
bushels  would  that  be  apiece  ? 

10.  If  20|-  dollars  be  paid  for  15  days'  work,  how  much 
would  that  be  per  day  ? 

11.  How  far  per  hour  is  88^  miles  in  17  hours? 

12.  How  far  per  day  is  476f  miles'  travel  in  8  days  ? 

13.  If  15  men  divide  among  themselves  77^  barrels  of 
apples,  what  would  be  the  share  of  each  man  ? 

14.  If  23  yards  of  cloth  cost  152f  dollars,  what  would  that 
be  a  yard  ? 

15.  How  much  is  the  cost  of  1  yard  of  cotton  cloth,  when 
3jVtt  dollars  are  given  for  35  yards  ? 

16.  How  many  times  is  25  contained  in  59f  ? 

17.  What  is  -fV  (93)  of  148^? 

18.  Divide  5|  by  12? 


FRACTIONS.  83 

19.  How  many  times  is  9  contained. in  47  ^^  ? 

20.  What  is  i  of  H? 

21.  Whatis^of2|? 

22.  What  is  i  of  18f  ? 

23.  Divide  4f  by  5. 

24.  Divide  1731f  by  12. 

25.  Divide  65542^^  by  256. 
|j|j26.  Divide  16388^  by  128. 

Ul4.   Model  of  a  Recitation. 

1.  If  the  postage  of  4  letters,  between  the  same  towns,  be 
I  of  a  dollar,  how  much  would  that  be  apiece  ? 

Since  1  letter  is  J  of  4  letters,  the  postage  of  1  letter  must 
be  J  of  the  postage  of  4  letters. 

|_-^  =-j?^  of  a  dollar.  Therefore,  as  the  postage  of 
4  letters  is  |  of  a  dollar,  the 
postage  of  1  letter  would  be  J  of  |  of  a  dollar.  But  the  nurri' 
her  of  parts  not  having  4  for  a  factor,  you  must  perform  the 
division  upon  the  size  of  the  parts,  which  you  can  do  by 
multiplying  thP  denominator  by  the  divisor,  retaining  the 
same  numerator. 

It  is  evident  that,  by  multiplying  the  denjominator  by  4, 
the  parts  are  made  \  as  large,  from  the  fact  that,  it  will  take 
4  times  as  many  of  them  to  make  the  unit  as  before.  It  takes 
only  ^fourths  of  a  dollar  (4  quarters  of  a  dollar)  to  make  a 
dollar ;  whereas  it  takes  4  times  as  many,  or  16  sixteenths  of  a 
dollar,  (16  fourpence-half- pennies,)  to  make  a  dollar. 

115>,   Observation. 

Observe,  (ll^j)  that  by  whatsoever  number  the  denomi- 
nator of  a  fraction  be  multiplied,  retaining  the  savne 
numerator,  the  fraction  is  thus  divided  by  that  number  ; 
for  the  denominator  showing  the  number  of  parts  that  make  the 
unit,  (83^)  their  size  is  diminished  in  the  same  ratio  that 

THE  denominator  IS  INCREASED. 

Observe,  also,  that  the  division  of  a  fraction  may  be  per- 
formed, either  upon  the  number  of  the  par^fs,  their  size  remain- 
ing the  same,  (IIO5)  or  upon  the.  size  of  the  parts,  their 
number  remaining  the  same,  (1 14: 5)  hut,  that  the  former  pro- 
cess is  to  be  adopted  in  all  cases  ivhen  the  divisor  is  a  factor  of 
the  numerator^  because  it  will  give  the  result  i7i  lower  terms. 


I 


84  arithmetic. 

116.    Exercises   in   Dividing  a  Fraction  by  Multiply- 
ing ITS  Denominator. 

In  like  manner^  solve  and  explain  the  following  problems. 

1.  If  2  boys  having  J  of  a  melon  divide  it  equally  between 
themselves,  what  would  be  the  share  of  each  ? 

2.  What  is  i  of  I  ? 

3.  Suppose  I  of  a  pie  to  be  cut  into  2  equal  pieces,  what 
part  of  the  whole  pie  would  each  piece  be?  What  is  \ 
ofi? 

4.  A  boy  having  f  of  a  dollar,  gave  \  of  it  for  a  pen- 
knife. What  part  of  a  dollar  did  his  knife  cost  ?  What  is  \ 
of  I  ? 

5.  If  2  shillings  are  J  of  ar  dollar,  what  part  of  a  dollar  is 
1  shilling? 

6.  If  a  boy  having  -5^  of  a  pie  should  give  \  of  it  to  his 
sister,  what  part  of  a  pie  would  he  give  away,  and  what  part 
would  he  keep  ? 

7.  If  ^  of  a  dollar  be  divided  equally  among  3  boys,  what 
part  of  a  dollar  is  the  share  of  each  ? 

8.  If  you  should  make  a  circle  on  your  slate,  and  draw  a 
line  across  it  through  the  centre,  how  many  parts  would  you 
make  of  it  ?     What  would  be  the  name  of  each  part  ? 

9.  If  from  the  centre  of  said  circle  you  draw  a  line  through 
the  middle  of  one  half,  making  two  parts  of  that  half,  how 
many  such  parts  would  make  the  whole  circle  ?  What  is  the 
name  for  such  parts  ?     What  is  ^  of  ^  ? 

10.  If  from  the  centre  of  said  circle  you  draw  a  line 
through  the  middle  of  one  fourth,  making  two  parts  of  that 
fourth,  how  many  such  parts  would  make  the  whole  circle  ? 
What  is  the  name  for  such  parts?     What  is  4  of  J  ? 

11.  What  is  1  of  ^  ?     W^hat  is  |  of  yV  ? 

12.  If  3  pounds  of  butter  cost  §  of  a  dollar,  what  is  that  a 
pound  ? 

13.  At  I  cf  a  dollar  for  4  bushels  of  apples,  what  would 
be  the  cost  of  a  bushel  ? 

14.  At  I  of  a  dollar  for  7  gallons  of  vinegar,  what  would 
be  the  cost  of  a  gallon  ? 

15.  If  6  bushels  of  wheat  cost  4|-  dollars,  what  would  that 
be  a  bushel  ? 

16.  If  4  dollars  buy  5§  bushels  of  rye,  how  much  would 
one  dollar  buy  ? 


^K  FRACTIONS.  85 

i   %   If  4  dollars  buy  3J  yards  of  silk,  how  much  might  be 
»ught  for  1  dollar  ? 

18.  If  18  pounds  of  raisins  cost  2f  dollars,  what  is  that  a 
pound? 

19.  If  16  hats  cost  48|  dollars,  what  would  1  hat  cost  ? 

20.  What  would  1  yard  of  broadcloth  cost,  if  25  yards 
ccst  150|  dollars? 

21.  How  far  per  hour  would  a  train  of  cars  go,  if  it  run 
l'!5J  miles  in  7  hours  ? 

22.  Divide  24|  by  7. 

23.  What  part  (87)  of  5  is  2? 

24.  What  part  of  5  is  2i  ? 

25.  What  part  of  5  is  l|  ? 

26.  What  part  of  12  is  7  ? 
'27.    What  part  of  12  is  3^\  ? 

117.   Illustration  of  the  Principle  of  Dividing  by  the 
Factors  of  the  Divisor. 

By  multiplying  one  number  by  another,  we  introduce  ipto 
the  multiplicand  all  the  factors  composing  the  multiplier, 
(Sdj)  and  the  product  will  be  composed  of  all  the  factors  of 
both  multiplier  and  multiplicand.  Also,  in  dividing  one 
number  by  another,  we  take  from  the  dividend  all  the  factors 
composing  the  divisor,  (755)  and  the  quotient  will  be  com- 
posed of  all  those  factors  composing  the  dividend,  which  are 
not  necessary  to  compose  the  divisor.  Thus,  by  multiplying 
35  =  7  X  5  by  33  =  3  X  H,  vve  obtain  1155  =  7  X  5  X  3 
X  11.  Now,  by  dividing  1155  =  7  X  5  X  3  X  H  by  105 
=  7  X  5  X  3,  the  quotient  is  the  remaining  factor,  11 ;  or, 
if  we  divi4^-by  35  =  7  X  5  the  quotient  is  33  =  3  X  H.  the 
remaining  two  factors  ;  or,  if  we  divide  by  7,  the  quotient  is 
165  =  5  X  3  X  11)  the  product  of  the  remaining  three  factors. 

Consequently,  by  dividing  the  product  of  several  factors  by 
some  of  them,  the  quotient  will  be  the  product  of  the  others. 

Also,  when  convenient,  we  may  separate  a  divisor  into 
factors,  and  take  them  from  the  dividend,  one  at  a  time,  that 
is.,  divide  first  by  one  factor,  then  divide  the  quotient,  thus 
obtained,  by  another  factor,  and  so  on,  wfth  all  the  factors  of 
the  divisor ;  the  last  .quotient  will  be  the  quotient  required. 
Thus,  instead  of  dividing  1155  directly  by  21,  we  may 
divide  first  by  7,  obtaining  165,  which  divided  by  3  gives  55, 
the  true  quotient. 
8 


96  arithmetic. 

118.  Model  of  a  Recitation. 

1.  Divide  875  by  35. 

7  )  875  '^^^  quotient  is  ^\  of  the  dividend.     Divide 

' first  by  7,  one  of  the  factors  of  the  divisor,  to 

5 )  125         obtain  j-  of  the  dividend,  and  then  divide  the 

' quotient  thus  obtained,  by  5,  the  other  factor 

2^         of  the  divisor,  to  obtain  -J  of  -f ,  or  -^  of  the 
dividend,  as  required. 

2.  Divide  3|  by  36. 

Since  36  is  not  a  factor  of  the  numerator,  but  4,  one  of  the 

factors  of  36,  is  also  a  factor  of  the  numer- 

3^  =  -^.  ator,  divide  first  by  4,  by  taking  J  as  many 

parts,  (lllj)  and  then  divide  that  quotient 

■^5^~5^"  =  ^?-       %  ^'  ^^^  other  factor  of  36,  by  making 

the  parts  ^  as  large,  (114,)  which  will 

give  i  of  J,  or  -rf^  of -^,  equal  to  ^,  which   is  the   true 

quotient  required. 

119.  Exercises   in   Dividing   by   the    Factors   of   the 

Divisor. 

hi  like  manner,  solve  and  explain  the  following  problems, 

1.  Divide  1421  by  49. 

2.  How  many  times  is  72  contained  in  1728  ? 

3.  How  many  casks  of  63  gallons   each,  may  be  filled 
from  7875  gallons  ? 

4.  If  a  horse  travel  f  of  a  mile  in  12  minutes,  how  far 
would  he  travel  per  minute  ? 

5.  If  21  dollars  buy  3^  barrels  of  flour,  what  part  of  a 
barrel  would  1  dollar  buy  ? 

6.  How  manv  times  is  14  contained  in  72|  ? 

7.  Divide  12|  by  15. 

8.  Divide  108f  by  18. 

9.  How  many  times  is  30  contained  in  72|  ? 

10.  What  part  of  a  time  is  27  contained  in  3^^? 

11.  What  part  of  a  time  is  28  contained  in  8|  ? 

12.  How  many  times  is  36  contained  in  42-^? 

13.  Divide  175  by  21. 

14.  Divide  1836  by  24. 

15.  What  is  the  quotient  of  960  divided  by  45  ^ 

16.  Divide  2^-2^  by  24. 

17.  Divide  38f^  by  36. 


FRACTIONS. 


87 


18.  Divide  3492  by  81. 

19.  What  part  of  15  is  10  ? 

20.  What  part  of  18  is  5f  ? 

21.  What  part  of  21  is  4J  ? 

22.  If  8512  be  the  product  of  three  factors,  two  of  which 
aie  8  and  19,  what  is  the  third  factor  ? 

23.  If  17160  be  the  product  of  8,  11,  13,  and  two  other 
factors,  what  are  the  other  two  factors  ? 

24.  One  of  two  factors  composing  1625  is  25.  What  is 
the  other  ? 

25.  Divide  17  X  19  X  10  by  19. 

26.  Divide  12  X  14  X  9  X  6  by  12  X  6. 

27.  How  many  times  is  3x5x7  contained  in  9  X  10 
><  14? 

28.  What  is  the  quotient  of  16  X  39  X  i  divided  by  2  X 
7X  13? 

130.   Illustration  of  the  Principle  of  Reducing  a  Frac- 
tion TO  Other  Terms  of  Equal  Value. 

« 

Make  a  circle  on  your  slate,  and 
draw  a  line  across  it  through  its 
centre,  making  two  half-circles. 
From  the  centre  draw  a  line 
through  the  middle  of  one  of  these 
halves,  and  from  the  same  point 
draw  a  line  through  the  middle  of 
each  of  the  two  fourths  made  of 
this  half  by  the  last  line  ;  thus 
making  the  ^  =  f .  Now  erase 
the  three  lines  last  drawn ;  thus 
making  the  f  =  |  again. 

21.    Observation. 

Observe,  (120^)  that  both  terms  in  \  are  made  4  tiTues  as 
large  in  its  equal  fraction  | :  that  is^  both  terras  in  J  have 
been  multiplied  by  4 ;  thus  making  the  parts  4  times  as  many^ 
but  \  as  large  in  ^  as  in  ^,  ^ 

jSo,  multiplying  both  terms  of  a  fraction  by  any  number, 
will  reduce  it  to  an  equal  fraction  in  higher  terms,  For^ 
tohile  it  multiplies  the  fractio?i  by  increasing  the  number 
oj  parts,  (lOlj)  it  also  divides  it  by  diminishing  the  size 


86 


ARITHMETIC. 


of  the  parts  (114)  in  the  same  ratio  as  their  nwmber  is  in^ 
creased. 

Observe,  also^  that  both  terms  in  |  are  made  J  as  large 
in  its  eqvul  fraction  ^  ;  that  is,  both  terms  in  f  have  been 
divided  by  4 :  thus  making  the  parts  \  as  many,  but  4  times 
as  large  in  ^  as  in  %, 

So,  dividing  both  terms  of  a  fraction  by  any  number,  will 
reduce  it  to  an  equal  fraction  in  lower  terms.  For,  while 
it  divides  the  fraction,  by  diminishing  the  number  of  parts, 
(11  Ij)  it  also  multiplies  it  by  increasing  the  size  of  the 
PARTS  (106)  in  the  same  ratio  that  their  number  is  dimin^ 
ished. 

133.   Illustration  of  the  Mode  of  Reducing  a  Frac- 
tion TO  Lower  Terms. 

1.   Reduce  f  J  to  lower  terms. 

Since  3, 7,  and  21  are  factors  com- 
^)  ii  =  -^  f'^^^on  to  both  terms  of  the  fraction, 

7\2i-_.3\3  1         you  may  divide  both  terms  (131) 

/ft  )  r^       t         ^y  either  of  these  common  factors, 

^1 )  f  i  =  i  But  observe,  that,  the  larger  the 

factor  used,  the  lower  will  be  the 
terms  to  which  the  fraction  will  be  reduced ;  and  that,  by 
using  the  greatest  common  factor ,  the  fraction  will  be  reduced 
to  its  lowest  terms. 

When  any  other  than  the  greatest  common  factor  is  used, 
the  new  fraction  obtained  may  be  reduced  lower,  by  using 
some  other  common  factor. 

133.   Factoring  of  Numbers. 

In  small  numbers,  the  factors  and  common  factors  may  be 
ascertained  by  observation ;  but  in  larger  numbers  other  means 
become  necessary. 

A  multiple  of  a  number  is  a  number  of  which  the  former 
number  is  a  factor;  as  multiples  of  3  are  6,  9,  &c.,  of  which 
3  is  a  factor. 

A  common  multiple,  of  two  or  more  numbers,  is  a  number 
of  which  those  numbers  are  factors. 

A  number  composed  of  factors  is  a  multiple  of  any  one  of 
those  factors ;  and,  also,  of  any  combination  of  its  prime  fac- 
tors.    (39.) 

•  When  one  number  is  a  factor  of  another,  all  the  factors  of 
the  former  are  also  factors  of  the  latter.  Thus,  21  being  a 
factor  of  63,  7  and  3,  the  factors  of  21,  are  also  factors  of  63. 


FRACTIONS.  89 

And  they  should  be  ;  for  63,  being  3  times  21,  should  contain 
7  and  3  three  times  as  often  as  21  contains  them. 

Hence,  a  factor  of  a  number  is  a  factor  of  any  multiple  of 
t  lat  number. 

Any  common  factor  of  two  numbers  is  a  factor  of  their 
sum,  and  of  their  difference.  For,  each  of  the  numbers  con- 
taining the  common  factor  a  certain  number  of  times,  their 
sum  must  contain  it  as  many  times  as  both  of  the  numbers  ; 
End  their  difference  must  contain  it  as  many  times  as  the 
krger  of  the  numbers  contains  it  times  more  than  the  smaller. 

Thus,  4,  being  a  common  factor  of  12  and  20,  must  be  a 
factor  of  their  sum,  and  of  their  difference  :  for  12  is  3  fours, 
E.nd  20  is  5  fours  ;  their  sum  will  be  5  fours  -j-  ^  fours  =  8 
fours,  or  32 ;  and  their  difference  will  be,  5  fours  —  3  fours 
==  2  fours,  or  8. 

2  is  a  factor  of  every  even  number. 

Any  number  ending  with  a  cipher  (10)  is  a  muUiple  oi 
".0,  consequently,  10,  and  the  factors  of  10,  are  factors  of  it. 

Any  number  ending  with  two  ciphers  (10)  is  a  multiple  of 
.00,  consequently,  100,  and  the  factors  of  100,  are  factors  of  it. 

5  is  a  factor  of  any  number  ending  with  5 ;  for  all  of  the 
number  but  5  is  a  multiple  of  10  of  which  5  is  a  factor. 

Any  factors  of  the  last  two  figures  of  a  number,  which  are 
also  factors  of  100,  are  factors  of  the  whole  number ;  for  all 
of  the  number  but  these  two  figures  is  a  multiply  of  100. 

8,  being  a  factor  of  200,  will  be  a  factor  of  any  number 
which  has  even  hundreds,  if  it  be  a  factor  of  the  last  two 
figures  of  the  number. 

9  is  a  factor  of  any  number,  when  it  is  a  factor  of  the  sum 
of  the  digits  which  express  that  number.  For  the  excess  in 
the  value  expressed  by  the  digits,  above  what  they  would  ex- 
press in  the  units'  place,  is  a  multiple  of  9  ;  since  every  re- 
moval of  a  figure  one  degree  higher  causes  that  figure  to 
express  teii  times  its  former  value,  (4,)  it  gains  by  each 
removal  9  times  the  value  it  would  have  before  the  removal. 

Thus,  10  is  9  more  than  1  ;  and  100  is  9  X  10  =  90 
more  than  10,  or  99  more  than  1,  &c.  Also,  70  is  9  X  7 
=  63  more  than  7  ;  and  700  is  9  X  70  =  630  more  than 
70,  or  9  X  70  +  9  X  7  =  693  more  than  7. 

Also  3,  being  a  factor  of  any  multiple  of  9,  is  a  factor  of 
any  number  when  it  is  a  factor  of  the  sum  of  the  digits  which 
ei^press  that  number. 


90  ARITHMETIC. 

Every  factor  of  a  number  has  its  corresponding  factor, 
which,  together,  compose  the  number,  (75.) 

Hence,  to  find  the  prime  factors  of  a  number,  separate  it 
into  two  factors,  by  dividing  it  by  any  known  factor,  and 
proceed  in  the  same  manner  with  each  of  these  factors,  and 
so  on,  till  the  prime  factors  are  all  obtained. 

134.   Model  of  a  Recitation. 

Find  the  prime  factors  of  1296. 

Here  observe^  that  12  is  a  factor,  the  factors  of  which  are 
3,  2,  2.  The  quotient  of  1296  divided  by  12  is  lOS,  whose 
factors  are  12,  the  factors  of  which  are  3,  2,  2 ;  and  9,  the 
factors  of  which  are  3,  3.  In  all,  3,  2,  2  X  3,  2,  2  X  3,  3 ; 
or,  3,  3,  3,  3,  2,  2,  2,  2 ;  or  3^  2\  the  small  4  being  an  index 
to  show  the  number  of  factors  like  that  over  which  it  is 
placed,  (364.) 

1S5,   Exercises  in  Factoring  Numbers. 

In  like  manirher^  solve  and  explain  the  following  problems. 

Find  the  prime  factors  of  the  following  numbers.  72,  88, 
120,  612,  336,  648,  930,  924,  936,  450,  360,  966,  870,  684, 
396,  432,  2480,  8000,  10449,  10503,  24876. 

136.   Model  of  a  Recitatioti. 

Reduce  %%  to  an  equal  fraction  in  its  lowest  terms. 

Take  -^t^  as  many  parts,  (lll^)  and  make 
12 )  14  =  !••         them  12  times  as  large,  ( 106^}  which  gives 
f ,  the  answer  required. 

1S7.    Exercises  in  Reducing  Fractions  to  Lower  Terms. 

In  like  manner^  solve  and  explain  the  following  problems, 

1.  Reduce  -^  to  its  lowest  terms. 

2.  Reduce  -/_,  |2^  ^4^  ^a,  J5^  i|  ^nd  ^|  to  their  lowest 
terms. 

3.  Reduce  J|,  ^^-,  f |,  f|,  -^^j,  and  -f^  to  their  lowest 
terms. 

4.  Reduce  -^^j,  f |^,  -^i^-^  and  Jfg  to  their  lowest  terms. 

5.  Reduce  /^o^,  J^Vf^  Hf  ^nd  yVV?  to  their  lowest 
terms. 

6.  Reduce  yVsV  ^^  ^^^  lowest  terms. 


FRACTIONS.  91 

128»   Model  of  a  Recitation. 

It  is  often  most  convenient,  in  reducing  a  fraction  to  its 
lowest  terms,  to  make  use  of  the  greatest  common  factor  of 
its  terms,  (122.)  Hence  it  will  be  useful  to  find  a  more 
d  rect  way  by  which  the  greatest  common  factor  may  always 
b'i  ascertained. 

Reduce  ^^^  to  its  lowest  terms. 

The  greatest  common  factor,  being  a 

61)  14S  (2  factor  of  148  and  of  64,  consequently, 

128  (123.)  of  64  X  2=  128,  will  be  a 

factor   of  148  —  128  =  20,  (123.) 

20)64  (3  Again,  this  factor,  being  a  factor  of 

60  ^         64  and  of  20,  consequently,  (123,) 

—  of  20  X  3  =  60,  will  be  a  factor  of 

4)20(5       64  —  60  =  4,  (123,)     But  4,  being 

20  a  factor  of  20  and  of  itself,  (123,) 

—  will  be  a  factor  of  20  X  3  +  4  =  64, 

-     g^  ig  one  of  the  given  numbers.     Again,  4, 

^ )  Tf  B  —  Tt •  being  a  factor  of  64  and  of  20,  will 

be  a  factor   (123)   of  64  X  2  +  20 
==  148,  the  other  given  number. 

Hence,  as  4  contains  the  greatest  common  factor  of  the 
given  numbers,  and  is  a  factor  of  therny  it  must  be  their 
greatest  common  factor.  Therefore,  take  \  as  many  parts, 
and  make  them  4  times  as  large,  which  gives  -Jf ,  the  answer 
required. 

129.    Observation. 

Observe,  (128^)  that^  the  greater  of  two  given  numbers 
being  divided  by  the  less,  the  less  by  the  first  remainder,  the 
first  remainder  by  the  second,  the  second  by  the  third,  <^c,,  till 
there  be  no  remainder  ;  the  greatest  common  factor  of  the 
given  numbers  will  be  a  factor  of  the  several  remainders  ;  for 
the  remainders  are  differences  (123)  between  numbers  of 
lohich  this  greatest  common  factor  is  a  factor.  Consequently, 
the  greatest  common  factor  of  the  given  numbers  cannot  exceed 

the    last    remainder.       But   THE    LAST    REMAINDER    IS    ITSELF 

THAT  FACTOR ;  for,  retracing  the  several  remainders  and  given 
numbers  from  the  lost  remainder  to  the  larger  given  number, 
observe  tJiat  the  last  remainder  is  a  factor  of  the  next  precede 


92  ARITHMETIC. 

ing ;  that  each  of  them  ^  added  to  the  next  preceding,  or  a 
multiple  of  it,  makes  the  next  in  order  ;  and  that,  therefore, 
the  lojSt  remainder  must  he  a  factor  of  them  all,  (I$J3  2)  hence, 
as  the  last  remaindei ,  both  contains  the  greatest  common 
FACTOR  of  the  given  numbers,  and  is  a  factor  of  them,  it 
must  he  their  greatest  common  factor. 

130*    Exercises  in  reducing  Fractions  to  their  Lowest 
Terms  by  the  Greatest  Common  Factor. 

In  like  manner,  solve  and  explain  the  folloioing  prohlems, 

1.  Ascertain  the  greatest  common  factor  of  30  and  72. 

2.  Reduce  ^^   to  its  lowest  terms. 

3.  Ascertain  the  greatest  common  factor  of  126  and  342. 

4.  Reduce  -Jj-|-  to  its  lowest  terms 

•5.  Ascertain  the  greatest  common  factor  of  128  and  176. 

6.  Reduce  ^f  to  its  lowest  terms. 

7.  Reduce  -ff f  to  its  lowest  terms. 

8.  Reduce  ^/^  to  its  lowest  terms. 

9.  Reduce  ^^|-  to  its  lowest  terms. 

10.  What  is  the  greatest  common  factor  of  384  and  1152  ? 

11.  What  are  the  lowest  terms  of  ^^^? 

12.  What  is  the  greatest  common  factor  of  114  and  285  ? 

13.  Reduce  ^^^  to  its  lowest  terms. 

14.  Reduce  ^f  f  f  to  its  lowest  terms. 

15.  What  are  the  lowest  terms  of  7^  ? 

16.  What  are  the  lowest  terms  of  ylf^  ? 

17.  Reduce  -^-f^is  to  its  lowest  terms. 

131.   Illustration  of  the  Least   Common  Multiple  op 
Numbers. 

2  is  a  factor  of  4,  6,  8,  10,  12,  14,  16,  18,  &c., 
and  3  is  a  factor  of      6,     9,      12,      15,     18,  &c. ; 
consequently,  4,  6,  8,  10,  &c.,  are  multiples  of  2;   and  6,  9, 
12,  &c.,  are   multiples  of  3.     But  6,  12,  18,  &c.,  are  mul- 
tiples of  hoth  2  and  3 ;  hence,  they  are  common  multiples  of 
2  and  3 ;  and  6  is  the  least  common  multiple  of  2  and  3. 

133.    Illustration  of  the  Least  Common    Denominator 
OF  Fractions. 

1.    Reduce  \,  also  \,  to  equal  fractions  in  higher  terms. 


I 


^^1 


FRACTIONS.  93 


multiplying  both  terms  of  each  fraction  by  2,  3,  4,  5, 
♦kc.,  successively, 

>  becomes  < 
4  )  (  f    ==    f    =    A-    =   A    =   f?>  &c. 

Observe,  that  the  denominators  of  the  fractions  to  which 
\  may  be  reduced,  will  be  multiples  of  2,  the  denjominator  of 
\ ;  and  that  the  denominators  of  the  fractiAms  to  which  \  may 
he  reduced^  will  be  multiples  of^,  the  denominator  of\. 

But,  PARTICULARLY  OBSERVE,  that  the  COMMON  MULTIPLES  of 

2  and  3,  the  denominators  of  \  and  \,  may  be  common  de- 
nominators of  fractions  to  which  \  and  \  may  be  reduced  ; 
%nd  that  the  least  common  multiple  of  the  denominators  of\ 
2nd  \  will  be  the  least  common  denominator  to  which 
I  and  J  can  be  reduced, 

133.  Model  of  a  Recitation. 

2.  Reduce  f  and  J  to  equal  fractions  having  their  least 
common  denominator. 

By  multiplying  both   terms   of 
|=:^  =  -5;f  =  f  J        each  fraction  by  2,  3,  4,  &c.,  suc- 
cessively, you  obtain  for  denomina- 
f    =    -^^    =    j^         tors  all  the  multiples  of  the  given 
denominators  as  far  as  you  proceed ; 
consequently,  the  first  common  denominator  thus  obtained, 
will  be  the  least  common  denominator  of  the  given  fractions. 

I134L*  Exercises  in  Reducing  Fractions  to  their  Least 
^  Common  Denominator. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  Reduce  f  and  |  to  equal  fractions  having  their  least 
common  denominator 

2.  Reduce  \  and  f  to  their  least  common  denominator. 

3.  Reduce  %  and  \  to  their  least  common  denominator.       ' 

4.  Reduce  \  and  f-  to  their  least  common  denominator. 

5.  Reduce  f  and  f  to  their  least  common  denominator. 

6.  Reduce  |  and  f  to  their  least  coihmon  denominator. 

7.  .  Reduce  |  and  f  to  their  least  common  denominator. 

8.  Reduce  f  and  I-  to  their  least  common  denominator. 

9.  Reduce  \  and  -^^j  to  their  least  common  denominator. 


94  ARITHMETIC. 

10.  Keduce  f  and  -^^  to  their  least  common  denominator. 

11.  Reduce  ^jj  and  /g-  to  their  least  common  denominator. 

12.  Reduce  ■^'^  and  -f^  to  their  least  common  denominator. 

13.  Reduce  -^  and  f^  to  their  least  common  denominator. 

14.  Reduce  -fg  and  fy  to  their  least  common  denominator. 

13S»   Mode  of  finding  the  Least  Common  Multiple. 

If  you  know  the  right  numbers  by  which  to  multiply  both 
terms  of  each  fraction,  to  reduce  the  fractions  to  their  least 
common  denominator,  only  one  multiplication  for  each  fraction 
would  be  necessary. 

Hence,  as  you  will  often  have  occasion  to  reduce  fractions 
to  their  least  common  denominator,  it  is  desirable  to  find  a 
more  direct  way  to  ascertain  the  right  multipliers. 

Every  number  which  is  not  a  prime  number,  is  composed 
of  prime  factors,  (29.)     Thus  :   24  =  3x2x2x2. 

Though  4,  6,  8  and  12  are  factors  of  24,  yet  they  them- 
selves are  composed  of  prime  factors,  and,  therefore,  are  com- 
posite factors, 

A  multiple,  or  composite  number,  is  composed  of  exactly 
all  its  prime  factors.  Hence,  a  number  which  contains  the 
prime  factors  of  another  number,  is  a  multiple  of  that  other 
number  ;  also,  a  number  which  contains  the  prime  factors  of 
two,  or  more  other  numbers,  is  a  common  multiple  of  those 
other  numbers. 

Consequently,  the  least  common  multiple  of  two  or  more 
giv€7inumbers,will  be  composed  of  suck  of  their  prime  factors, 
and  only  such,  as  are  necessary  to  compose  each  of  the  given 
numbers. 

Thus:  6  =  3x2,  and8=:2x2x2;  now  take  3x2, 
the  factors  of  6,  and  2x2,  the  factors  which  8  has  that  6 
has  not,  and  you  have  all  the  factors  of  6  and  8 ;  viz : 
3  X  2  X  2  X  2=: 24,  the  leasi  common  multiple  of  6  and  8. 

Hence,  to  ascertain  the  least  comrnon  multiple  of  two  or 
more  given  numbers,  it  is  only  necessary  to  separate  the 
given  numbers  into  their  prime  factors,  and  to  select  and 
multiply  together  such,  and  only  such  of  the  factors  as  are 
necessary  to  compose  each  of  the  given  numbers, 

130*   Model  of  a  Recitation. 

1.   Ascertain  the  least  common  multiple  of  14  and  21. 


I 


FRACTIONS.  95 


Ij^ o  v7  ^^  iskes  2  and  7  to  compose  14, 

21  ZI Q  V  7'  ^^^  ^^^^  *^^^  ''''  ^^o®^^^^  with  the 

0  sT^  V ?9  '    40  3,  compose  21 ;  therefore,  the  other 

^X  ^  X  cJ  —  ^.  ^  ^^.^^  omitted,   2  X  7  X  3  =  42, 

'vrill  be  the  least  common  multiple  required. 

]l37*   Mode  of  reducing  Fractions  to  their  Least  Com- 
mon Denominator. 
* 

2.  Reduce  -^^  and-^T  to  equal  fractions  having  their  least 
common  denominator. 

Since  the  least  common  denominator  will  be  the  least  com- 
mon multiple  of  the  given 
Ti  =  ^  =  YKTh='T^'  denominators,    (ISS,)    it 

will   only  be  necessary  to 
^  =  ^  =  ^f^2-  =  :fV  separate  the  given  denom- 

inators into  their  prime 
factors,  and  multiply  both  terms  of  each  fraction  by  such 
factors  composing  the  denominator  of  the  other  fraction, 
as  are  necessary  to  make  each  denominator  equal  to  the  least 
common  multiple  of  the  given  denominators ;  that  is,  multi- 
ply both  terms  of  each  fraction  by  the  factors,  composing  the 
denominator  of  the  other  fraction,  which  it  has  not  already  in 
its  own  denominator. 

Thus ;  by  multiplying  both  terms  of  the  first  fraction 
by  3,  and  of  the  second  by  2,  the  denominators  will  be  com- 
posed of  the  same  factors,  and  only  such  as  are  indispensa- 
ble ;  consequently,  the  fractions  are  reduced  to  equal  fractions 
having  their  least  common  denominator  as  required. 

138.   Model  of  a  Recitation. 

3.  Reduce  ^  and  ^^  to  equal  fractions  having  their  least 
common  denominator. 

First,  reduce  the  frS.ctions  to 
^®^  =  I  =  ^1^  =  ^^..  their  lowest  terms,  then  separate 

the  denominators  into  their  prime 

-J-f  ==  yV  ==  :f  1; 3- =  i-J-  factors,    or,^  since  they  have  a 

common  composite  factor,  4,  this 

need  not  be  reduced  to  prime  factors ;  and,  finally,  multiply 

both  terms  of  the  first  fraction  by  3,  and  of  the  second  by 

2f  and  the  fraction  will  be  reduced  as  required. 


96  ARITHMETIC. 

4.  Reduce  ^,  ^,  and  ^5-,  to  their  least  common  denomi- 
nator. 

Multiply  both  terms   of  the   first 
|-  =  ^^Tj  =  §^.  fraction  by  5,  of  the  second  by  3, 

^  3    g  and  of  the  third  by  2,  and  the  several 

1^       ^x's       UTF*  denominators   will  be  composed  of 

-^^  =  -j4-^  =  -^jj,  the  same  factors  ;  consequently,  the 

given  fractions    will  be  reduced   to 
their  least  common  denominator  as  required.  , 

1.39.    Observation. 

Observe,  that,  to  reduce  two  or  more  fractions  to  their 
least  common  denominator^  we  first  reduce  the  fractions  to 
their  loioest  terms ;  second,  separate  these  denominators  into 
their  prime  factors  ;  and  third,  multiply  both  terms  of  each 
of  these  fractions  by  the  factors  belonging  to  the  other  de^ 
nominators  lohich  do  not  belong  to  its  own  denominator. 

140«    Exercises  in  reducing  Fractions  to  their  Least 
Common  Denominator. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  Ascertain  the  least  common  multiple  of  8  and  12. 

2.  Reduce  f  and  y^^  to  their  least  common  denominator. 

3.  Ascertain  the  least  common  multiple  of  8  and  14. 

4.  Reduce  f  and  -f-^  to  their  least  common  denominator. 

5.  Ascertain  the  least  common  multiple  of  9  and  15. 

6.  Reduce  f  and  -^-^  to  their  least  common  denominator. 

7.  Ascertain  the  least  common  multiple  of  15  and  18. 

8.  Reduce  -f-^  and  -f^  to  their  least  common  denomina- 
tor. 

9.  Ascertain  the  least  common  multiple  of  5  and  7. 

10.  Reduce  f  and  f  to  their  least  common  denominator. 

11.  Ascertain  the  least  common  multiple  of  2,  3,  5, 
and  7. 

12.  Reduce  \,  \,  \,  and  f,  to  their  least  common  denomi- 
nator. 

13.  Ascertain  the  least  common  multiple  of  10,  14, 
and  15. 

14.  Reduce  -^j,  y\-,  and  -^  to  their  least  common  denomi- 
nator. 

15.  Ascertain  the  least  common  multiple  of  250  and  400. 


w 


FRACTIONS.  97 

Reduce  -^^  and  -^^  to  their  least  common  denomi- 
nator. 

17.  Ascertain  the  least  common  multiple  of  15,  24  and 
'15. 

18.  Reduce  •^,  ^-j?  Q^^^d,-^^,  to  their  least  common  de- 
nominator. 

19.  Reduce  J-f-  and  -f^  to  their  least  common  denominator. 

20.  Reduce  -^  and  ^^  to  their  least  common  denominator, 

21.  Reduce  J^,  f|,  and  f|,  to  their  least  common  denom- 
iaator. 

22.  Reduce  ^  and  -^5-  to  their  least  common  denominator. 

23.  Reduce  y\,  -^j,  and  ^^,  to  their  least  common  de- 
r.ominator. 

24.  Reduce  -1^5^(5-  and  yf f§iy  to  their  least  common  de- 
rominator. 

1141 .   Model  of  a  Recitation. 

1 .  John  paid  f  of  a  dollar  ( 5  ninepences)  for  a  reading 
book,  J  of  a  dollar  for  a  writing  book,  and  j  of  a  dollar  for 
an  arithmetic ;  how  many  dollars  did  they  all  cost  ? 

Since  the  parts  expressed  by 
■^^^■Z.  =  -1^3-  =  1|  dolls.       these  several  fractions   are  all 

eighths^  and  since  the  numerator 
of  each  fraction  shows  the  number  of  parts  expressed  by  that 
fraction,  (83^)  the  sum  of  the  numerators  will  show  the 
number  of  parts  expressed  by  all  of  the  fractions  ;  therefore, 
place  the  sum  of  the  numerators  over  their  common  denomi- 
nator, and  the  result  will  be  the  sum  of  the  fractions,  as 
required. 

2.  If  I  pay  2|  dollars  for  a  pair  of  shoes,  and  4f  dollars 
for  a  pair  of  boots,  what  is  the  whole  cost  ? 

Here  are  3  parts  and  5  parts  making 
2|  =  2y^2-  S  parts,  but  they  are  all  neither  fourths^ 

4^-^410  nor  sixths  ;  if,  however,  you  reduce  the 

fractions    to   their    least    common  de- 

77    dollars        nominator,  ( I3O5)  the   parts    become 
^  ■      A  +  «  =  ^A^  =  +I=1t\-.   Write 

the  ^3^,  and  add  the  unit  with  the  other  units,  makmg  T^^g 
dollars,  which  is  the  answer  required. 


98  ARITHMETIC. 


143*   Model  of  a  Recitation. 

1.  A  boy,  having  f  of  a  dollar,  spent  f  of  a  dollar  for  a 
bunch  of  quills.     How  much  money  had  he  left  ? 

He  had  left  the  difference 

^    g 3 I     n      111  between  f  and  f .     Since  the 

w^       ^       t  •         parts  expressed  by  the  frac- 

tions are  all  sixths,  and  the 
numerators  show  the  number  of  the  parts,  the  difference  be- 
tween the  numerators  will  show  the  number  of  parts  he  had 
left,  which  continue  to  be  of  the  same  size  ;  therefore,  place  the 
difference  of  the  numerators  over  the  common  denominator^ 
and  the  result  will  be  the  difference  between  the  fractions,  as 
required. 

2.  If  a  man  earn  14|  dollars,  and  spend  4|-  dollars  in  a 
week,  what  would  he  save  in  a  week  ? 

He  would   save  the   difference 
j^i  ___  2^3  between  what  he  earned  and  what 

.5 .5  he  spent. 

%  -^    %  You  cannot  take  5  parts  from  3 

"~  parts  of  the  same  size  ;  therefore, 

9^  dollars.  reduce  1  of  the  14  units  to  sixths, 

(OSj)  making  6  sixths,  and  the  3 

sixths,  make  |,  from  which,  if  f  be  taken,  there  Avill  remain 

^-^  ==  |,  which  write  ;  and  then  take  4  units,  not  from  14 

units,  for  1  of  them  has  been  disposed  of,  but  take  4  units 

from  13  units,  and  there  will  remain  9  units  ;  making  9| 

dollars,  which  is  the  answer  required. 

14:3.    Exercises  in  adding  and  subtracting  Fractions. 

In  like  manner,  solve  and  explain  the  folloiving  problems. 

1.  If  you  buy  a  lead-pencil  for  -Jg-  of  a  dollar,  a  writing- 
book  for  -f^  of  a  dollar,  an  inkstand  for  y\  of  a  dollar,  how 
much  must  you  pay  for  the  whole  ? 

2.  At  a  contribution,  John  contributed  -^  of  a  dollar,  his 
brother  ^  of  a  dollar,  and  their  sister  -^^  of  a  dollar.  What 
did  they  all  contribute  ? 

3.  By  going  in  the  road,  John  walks  |-  of  a  mile  to  school, 
but  by  going  across  the  pastures  and  fields,  it  is  only  f  of  a 
mile  to  school.  How  much  can  he  save  in  distance  by  going 
the  nearer  way  ? 

4.  If  a  writing  book  cost  J  of  a  dollar,  and  a  quire  of  letter 


■ 


FRACTIONS.  99 


paper  cost  ^  of  a  dollar,  how  much  more  will  the  paper  cost 
t  lan  the  book  ? 

5.  If  Samuel  have  f  of  a  dollar,  and  Martin  have  §  of  a 
dollar,  how  much  have  both  of  them  ? 

6.  If  Isaac  have  |  of  a  dollar,  and  his  sister  f ,  which  has 
the  more  money,  and  how  much  more  than  the  other  ? 

7.  How  many  yards  of  cloth  in  4  pieces  which  measure 
as  follows,  18f  yards,  27|-  yards,  23|  yards,  and  25|  yards  ? 

8.  If  Mr.  Farmer  hire  2  men  and  a  boy  to  work  for  him 
a  week,  and  pay  them  as  follows,  5f  dollars  to  one  man, 
7 1  dollars  to  the  other  man,  and  3|  to  the  boy ;  how  much 
v^ould  he  pay  the  whole  ? 

9.  If  it  take  1 1  yards  of  cloth  to  make  a  coat,  and  |  of  a 
yard  to  make  a  pair  of  pantaloons,  how  much  more  cloth  in 
tlie  coat  than  in  the  pantaloons  ? 

10.  A  merchant  bought  a  piece  of  cloth,  containing  23 
yards,  and  sold  7f  yards  of  it.     How  much  of  it  had  he  left  ? 

11.  In  a  barrel  there  are  31 J  gallons,  and  in  a  hogshead 
63  gallons.  How  many  more  gallons  in  a  hogshead  than  in 
a  barrel ? 

12.  If  7 1  gallons  leak  out  of  a  barrel,  how  much  would 
remain  ?  « 

13.  John  works  ^  of  the  time,  plays  J  of  the  time,  sleeps 
J  of  the  time,  and  is  at  school  the  rest  of  the  time.  What 
part  of  the  time  is  he  at  school  ? 

14.  Of  the  road  that  John  walks  to  school,  J  is  up  hill,  J 
is  down  hill,  and  the  rest  is  level.  What  part  of  the  way  is 
level  road ;  and  how  much  more  of  the  way  is  up  hill  in 
going  to  school  than  in  returning  home  ? 

15.  A  pair  of  oxen  and  a  horse  compose  a  team  ;  one  ox 
draws  f  of  the  load,  the  other  ox  draws  |  of  the  load,  and  the 
horse  draws  the  rest  of  it.  How  much  more  do  the  oxen 
draw  than  the  horse  ? 

16.  Add  together  f  and  ^^. 

17.  What  is  the  sum,  and  difference  of  f  and  J  ? 

18.  Add  together  7|  and  lOf. 

19.  What  is  the  difference  between  13 5*5  and  17^  ? 

20.  What  is  the  sum,  and  difference  of  16^  and  12^\  ? 

21.  Subtract  24^3^  from  25t^. 

22.  How  much  more  is  12^^^  than  both  4f  and  5^^  ? 

23.  How  much  less  are  both  f  and  2 j  than  4  ? 

24.  How  much  more  is  the  sum  of  10/^  and  5^  than 
their  difference  ? 


100  ARITHMETIC. 

25.  How  much  is  1^--^-^  ? 

26.  How  much  more  is  ?2 +|j:i2  than  '1±^±^  ? 

27.  How  much  less  is  l^V'-^  than  ^-t^fi^  ? 

28.  How  much  are  i|--^5  and  i^?  ? 

29.  Add  together  ^,1,  J,  and  i 

30.  Add  together  ^\,  -if,  f f ,  and  f §. 

31.  Subtract  ^^\  from  f  f . 

144:«   Illustration  of  the  Principle  of  multiplying  by 
A  Fraction. 

At  4  dollars  a  yard  for  broad  cloth,  what  would  be  the  cost 

of  4  yards  ? — of  2  yards  ?  —  of  1  yard  ? — of  J  of  a  yard  ? — 

of  I  of  a  yard  ? 

If  1  yard  cost  4  dollars,  4  yards 

4  X  4  =  16  dollars.         would  cost  4  times  4  dollars,  equal 

A  Ky  o         Q  A  u  to  16  dollars.     2  yards  would  cost 

4  X  ^  —    »  dollars.         2  ^.^^^  ^  ^^^^^^^  ^^^^^  ^^  g  ^^^^^^^^ 

4x1=   4  dollars.         1  yard  would  cost  1  time  4  dollars, 

-       1 o  /I  11  equal   to   4   dollars.     |   of  a  yard 

<i  X  2  —    ^  aoiiars.         ^^^^^  ^^^^  ^  ^.^^  ^  dollars,  or,  more 

4  X  I  =  3  dollars.  properly,  J  of  4  dollars,  equal  to  2 
dollars.  |  of  |i  yard  would  cost  J 
time  4  dollars,  or,  more  properly,  J  of  4  dollars,  equal  to  1 
dollar  ;  but  |  of  a  yard  would  cost  3  times  as  much  as  J  of  a 
yard,  which  is  3  dollars. 

Observe,  that,  since  the  product  must  he  as  many  times  the 
multiplicand  as  there  are  units  in  the  multiplier,  (SAj) 
when  the  multiplier  is  1,  the  product  will  not  differ  from  the 
multiplicand,  when  the  multiplier  is  greater  than  1,  the  pro- 
duct will  be  greater  than  the  multiplicand ;  but  when  the 

multiplier  is  less  THAJ^  1,  THE  PRODUCT  WILL  BE  LESS  THAN 
THE  MULTIPLICAND,  AND  SUCH  A  PART  OF  THE  MULTIPLICAND  AS 
THE  MULTIPLIER  IS  OF  A  UNIT. 

14ftS.   Model  of  a  Recitation. 

1.  At  5  dollars  a  cord  for  wood,  what  would  be  the  cost  of 
2  cords  ?  —  of  I  of  a  cord  ? 

If  1   cord   costs  5 

r-    .ex       -lA  J  n  dollars,  2  cords  would 

5  X  2  =  10  dollars.  ^^^^  2  ^.^^^  5  ^^^^^^^^ 

^  .     o        ^^«        ,1V       o^  J  n  equal  to  10  dollars. 

6  X  i  =  H^^  =  V  =  3|  dollars.        ^^  ^^  ^  ^^^^  ^^^^ 

would  cost  f    of   5 


FRACTIONS.  101 

dc liars.     J  of  5  is  J,  (94,)  therefore,  |  of  5  will  be  3  times 
as  TTiany  fourths^  equal  to  -^,  =  3|  dollars. 

2.  At  7  dollars  a  barrel  for  flour,  what  would  be  the  cost 
of  3 1  barrels  ? 

If  1  barrel  costs 

7  X  ^=  -^  =^  =  25|  dollars.       Ltwoll^lot 

-y-  of  7  dollars. 
Since  ^  of  7  is  J,  -V"  of  7  will  be  11  times  J  =  zjoi  ^  j^ 
=:25|  dollars,  which  is  the  answer  required. 

Another  Explanation,  —  If  1  barrel  cost  7  dollars,  3 1  bar- 
rels would  cost  3|,  or  -y^,  times  as  much.  First,  multiply 
b^  11  as  if  it  were  11  units,  which  gives  77  dollars.  But,  as 
th?  right  multiplier  is  -y-,  only  ^  of  11  units,  (945)  the  right 
product  ought  to  be  only  ^  of  77  dollars  ;  therefore,  divide 
77  by  3,  which  gives  ^  =  25|  dollars,  (865)  as  before. 

1'16«    Observation. 

Observe,  (I455)  that,  in  multiplying  by  a  fraction,  the 
process  consists  of  two  steps,  on  account  of  the  tioo  numbers  in 
th^  multiplier ;  and,  that  either  step  may  he  taken  first,  prO' 
vided  the  reasoning  be  suited  to  the  process. 

147.    Exercises  in  multiplying  by  a  Fraction. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  If  a  man  earn  10  dollars  a  week,  how  much  would  he 
earn  in  |^  of  a  week  ? 

2.  If  you  can  walk  3  miles  an  hour,  how  far  can  you  walk 
in  I  of  an  hour  ? 

3.  If  board  be  3  dollars  a  week,  what  must  be  paid  for 
board  1^  weeks  ? 

4.  At  20  dollars  a  month,  what  is  a  man's  wages  3^^  of  a 
month  ? 

5.  If  3  dollars  buy  a  yard  of  cassimere,  what  must  be  paid 
for  2\  yards  ? 

6.  What  is  I  of  15  ? 

7.  Multiply  15  by  |. 

8.  If  a  barrel  of  mackerel  cost  8  dollars,  what  would  2| 
barrels  cost  ? 

9.  At  2  dollars  a  day,  what  would  be  the  wages  for  5^ 
days  ? 

9* 


102  ARITHMETIC. 


10. 

If  160  rods  make  an  acre,  how  many  rods  in  3f  acres  ? 

11. 

Multiply  10  by  5§. 

12. 

Multiply  5f  by  10. 

13. 

What  is  j\  of  16  ? 

14. 

Multiply  6  by  y\. 

15. 

Multiply  5  by  ^V- 

148.   Model  of  a  Recitation. 

1 .  At  f  of  a  dollar  per  day,  what  should  a  laborer  receive 
for  4  days'  work? — for  3  days'  work? — for  |  of  a  day's 
work  ?  —  for  J  of  a  day's  work  ? 

If  for  1  day's  work  he  re- 

q  q         /ii  J  11  ceive  |-  of  a  dollar,  for  4  days' 

t-z.-r=  #  =4i  dollars.  .    \       i,     u  •     "^  a 

*  •  *         '^  2  work    he    should    receive    4 

9^Ji  =  _2  7.  =  33  dollars  ^^"^^^  I  of  a  dollar;  which 

^  g  F  •  ascertain  by  making  the  parts 

9i^_3  ^r„  ^^]i_  4    times    as    large,   (106.) 

^       —  ^  ^^  ^  ^''^^^'^-  For  3  days'  work,  he  should 

9 g     f     ^  11  receive  3  times  f  of  a  dollar  ; 

^>^^       ^^  '  which  ascertain  by  making  3 

times  as  many  parts,  (101.) 
For  I  of  a  day's  work,  he  should  receive  ^  of  f  of  a  dollar ; 
which  ascertain  by  taking  |  as  many  parts,  (111.)  For  J 
of  a  day's  work,  he  should  receive  J  of  f  of  a  dollar ;  which 
ascertain  by  making  the  parts  J  as  large,  (114.) 

2.  If  a  horse  travel  6|  miles  per  hour,  how  far  would  he 
travel  in  4  hours  ? — in  |  of  an  hour  ? — in  5 J  hours  ? 

If  he  travel  6|  =  ^  miles  in  1 

?1_  __  27  miles.  hour,  in  4  hours  he  would  travel 

~  4  times  ^  of  a  mile  ;  which  ascer- 

27-^3 o Ai  ^:Uq  tain  by  making  the  parts  4  times 

4^2  —  f  —  ^2  ^i^es.         ^^  ^^^^^^  (106.)     In  I  of  an  hour 

27-^3X4       oc      -1  ^®  would  travel  |  of  ^-  of  a  mile. 

TT4—  =  ^^  ™^^^'  Divide  the  number  of  parts  (111) 

by  3,  to  obtain  J,  and  make  these 
parts  2  times  as  large,  (106^)  to  obtain  |,  which,  reduced, 
will  be  the  answer  required.  In  5J  =  J^  hours,  he  would 
travel  -^t^^-  of  ^-  of  a  mile.  Divide  the  number  of  parts  (111) 
by  3,  to  obtain  |,  and  multiply  the  quotient  by  16,  to  obtain 
^ ;  but,  since  4,  one  of  the  factors  of  16,  is  a  factor  of  the 
denominator,  (IO85)  multiply  by  4,  by  making  the  parts  4 
times  as  large,  and  then  multiply  by  4  again,  the  other  factor 


FRACTIONS.  103 

0 '  16,  by  making  4  times  as  many  parts,  which,  reduced,  will 
b3  the  answer  required.  In  reducing  this  expression  of  the 
a  iswer,  say :  3  in  27,  9  times,  and  4  times  9  are  36.  4  in  4, 
oice  ;  and,  since  the  denominator  is  1,  the  numerator,  36,  is 
units,  (83.) 

Another  Explanation.  —  If  he  travel  6|,  or  ^/-,  miles  in  1 
hour,  in  5\  hours  he  would  travel  5J,  or  -^,  times  as  far. 
r  irst,  multiply  by  16,  as  if  it  were  units,  which  gives  ^i^/ ; 
bit,  as  the  right  multiplier  is  J^,  only  \  of  16  units,* the 
right  product  ought  to  be  only  \  of  what  we  now  have  ;  there- 
f(tre,  divide  by  3,  which  gives  Y-V  "  ^  =  36,  as  before. 

149.   Observation. 

Observe,  (148,)  that^  in  multiplying  a  fraction  hy  a  frac- 
tion^ the  process  consists  of  two  steps  ^  either  of  which  may  be 
taken  first ;  that^  in  many  cases ^  there  are  two  loays  of  per- 
forming each  part  of  the  process^  on  account  of  the  two  numbers 
in  the  multiplicand^  but  that,  of  the  tivo  ways,  that  is  to  be 
adopted  which  loill  give  the  result  in  the  loioer  terms  ;  that  each 
p%rt  of  the  process  is  to  be  expressed  and  explained  separately  ; 
and  finally,  that  the  process  is  to  be  performed  by  reducing  the 
expression  of  the  result  to  its  simplest  terms. 

1€^*    Exercises  in  multiplying  Fractions  by  Fractions. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  If  a  benevolent  man,  having  only  ^  of  a  bushel  of 
wheat,  should  give  |  of  it  to  his  poor  neighbors,  what  part  of 
a  bushel  would  he  give  away? 

2.  At  y  of  a  dollar  a  yard,  what  part  of  a  dollar  would  -J 
of  a  yard  cost  ? 

3.  What  number  is  equal  to  it  ^^  nr-  ^ 

4.  If  a  yard  of  cloth  cost  5^  dollars,  what  would  |-  oi^  a 
yard  cost  ? 

5.  At  f  of  a  dollar  a  yard,  what  will  f  of  a  yard  cost  ? 

6.  At  I  of  a  dollar  a  pound,  what  will  J  of  a  pound  of  tea 
cost  ? 

7.  At  ^  of  a  dollar  a  pound,  what  will  f  of  a  pound  of 
coffee  cost  ? 

8.  At  2 J.  dollars  a  bushel,  what  will  6^  bushels  of  wheat 
cost  ? 

9.  At  /^  of  a  dollar  per  hour,  how  much  may  be  earned 
in  J  of  an  hour  ? 


104  ARITHMETIC. 

10.  At  6|  dollars  a  barrel,  what  will  ^^  of  a  barrel  of  flour 
cost  ? 

11.  If  7|  yards  of  satinet  be  bought  at  |  of  a  dollar  per 
yard,  what  would  be  the  whole  cost  ? 

12.  If  1  cord  of  wood  cost  6|  dollars,  what  will  7|  cords 
cost? 

13.  At  i^  of  a  dollar  a  pound,  what  will  17|  pounds  of 
sugar  cost  ? 

14.  At  3|-  shillings  a  yard,  what  will  8|  yards  of  ribbon 
cost  ? 

15.  If   1   dollar  buy  |  of  a  gallon  of  wnne,  how  much 
would  67^  dollars  buy  ? 

16.  What  is  the  value  of  36f  acres  of  land,  at  40^  dol- 
lars per  acre  ? 

17.  What  is  the  value  of  142J  tons  of  coal,  at  7f  dollars 
per  ton  ? 

18.  What  is  the  value  of  16|  tons  of  hay  at  11^  dollars 
per  ton  ? 

19.  What  will  7||  bushels  of  apples  cost  at  -JJ  of  a  dollar 
per  bushel  ? 

20.  A  merchant  owning  ^^  of  a  ship,  sold  f  of  his  share; 
what  part  of  the  whole  ship  did  he  sell  ? 

21.  What  is  f  of  I- ? 

22.  Multiply  ^^  by  f 

23.  Multiply  I  of  f  by  ^-3-. 

24.  What  is -fV  of  f  of  I  ? 

25.  What  is  ^^  of  ^f  multiplied  by  f  f  ? 

26.  Ascertain  the  product  of  the  following  factors,  J  X  § 

X  I  X  t- 

27.  How  much  is  f  of  |  of  |  of  |  ? 

28.  Multiply  8|  by  f 

29.  Multiply  ^\  by  173-3^. 
;30.   Multiply  IIH  by  SxV 

31.  What  is  the  second  power  (49)  of  |? 

32.  What  is  the  second  power  of  §  ? 

33.  What  is  the  third  power  of  |  ? 

34.  What  is  the  fourth  power  of  |  ? 

ISl.   Illustration  of  the   Principle  of   dividing  by  a 

Fraction. 

If  a  philanthropist  have  eight  dollars  to  distribute  to  the 
poor,  to  how  many  persons  could  he  give  4  dollars  apiece  ? 


FRACTIONS.  105 

2  dollars  apiece  ?    1  dollar  apiece  ?    ^  of  a  dollar  apiece  ? 
^  of  a  dollar  apiece  ? 

He  could  give  4  dollars  apiece  to  as  many  persons  as  there 

are  times  4  dollars  in  8  dollars. 
8  -r-  4  =    2  i^ersons.  He  could  give  2  dollars  apiece 

8  -=-  2  =    4  persons.         to  as  many  persons  as  there  are 
8  -T-  1  =   8  persons.         times  2  dollars  in  8  dollars. 
8  X  2  =  16  persons.  He  could  give  1  dollar  apiece 

8  X  4  =  32  persons.         to  as  many  persons  as  there  are 

times  1  dollar  in  8  dollars. 
He  could  give  J  of  a  dollar  apiece  to  as  many  persons 
as  there  are  times  |  of  a  dollar  in  8  dollars ;  and,  since  there 
are  2  halves  in  every  unit,  (S4L^)  there  will  be  2  times  as 
many  halves  as  units ;  therefore,  multiply  8  by  2  to  ascertain 
how  many  times  J  is  contained  in  it. 

He  could  give  |  of  a  dollar  apiece  to  as  many  persons  as 
there  are  times  J  of  a  dollar  in  8  dollars;  and,  since  there 
aie  f  in  a  unit,  there  will  be  4  tiqies  as  many  Js  as  units; 
therefore,  multiply  8  by  4  to  ascertain  how  many  times  J  is 
cc'ntained  in  it. 

Observe,  that,  since  the  divisor  shoivs  how  many  equal  parts, 
sixch  as  the  quotient,  will  make  the  dividend;  (63)  when  the 
divisor  is  1  the  quotient  will  not  differ  from  the  dividend  ; 
when  the  divisor  is  greater  than  1  the  qvjotient  will  he  less 
THAN  THE  DIVIDEND ;  hut  when  the  divisor  is  less  than  1  the 
quotient  will  be  greater  than  the  dividend. 

1^9*   Model  of  a  Recitation. 

1.  What  would  be  the  price  of  1  acre  of  land,  if  25  dollars 
be  paid  for  6  acres  ?  for  f  of  an  acre  ?  for  4f  acres  ? 

If  6  acres  be  bought,  paying  one  dollar  per  acre  would 

25  -  6  =  4i  dollars.  f  ^"''"t  ®  ^"^^^'^ '  *'''"'- 

^^  =  i^a  =  33i  dollars.         f"""^',  *«   ?"*=«   P^^  f"^ 
3  •*  "*  would  be  as  many  dollars 

as  6  dollars  is  contained  times  in  25  dollars. 

If  I  of  an  acre  be  bought,  paying  1  dollar  per  acre  would 
require  J  of  a  dollar;  therefore,  the  price  per  acre  would  be 
as  many  dollars  as  f  of  a  dollar  is  contained  times  in  25  dol- 
lars. Since  there  are  4  times  as  many  Js  as  units,  (08^)  in 
any  number,  multiply  by  4  to  ascertain  how  many  times  j  is 
contained;  then,  (since  f  is  3  times  as  much  as  J,  con- 
sequently, will  be  contained  only  J  as  often  as  J,)  divide  that 


106  ARITHMETIC. 

quotient  by  3  to  ascertain  how  many  times  f  is  contained, 
which  reduced  will  be  the  answer  required. 

If  4f  =.A^   of  an  acre  be  bought,  paying  1  dollar  per  acre 

would  require  -^  dollars  ; 

-f^Xa  =  .j.|  =  5^^  dollars.  therefore,    the    price   per 

acre  would  be  as  many 
dollars  as  -^  of  a  dollar  is  contained  times  in  25  dollars. 
Multiply  25  by  3  to  ascertain  how  many  times  J  is  con- 
tained; (99)  and,  since  ^  will  be  contained  i\  as  often, 
divide  that  quotient  by  14  to  ascertain  how  many  times  ^  is 
contained,  which  reduced  will  be  the  answer  required. 

2.  How  many  barrels  of  flour  could  a  trader  buy  for  48 
dollars,  at  6f  dollars  per  barrel  ? 

He  could  buy  as  many  barrels 

^8Xg£A=,^  =  7^         as  6f  ==-2^   of  a   dollar   is    con- 
tained times  in  48  dollars. 

Multiply  by  3  to  ascertain  how  many  times  J  is  contained 
in  48,  and  divide  that  quotient  by  20  to  ascertain  how  many 
times  ^-  is  contained ;  but,  since  4,  one  of  the  factors  of  20 
is  also  a  factor  of  48 ;  in  dividing  by  20,  first  divide  by  4, 
and  then  divide  that  quotient  by  5,  the  other  factor  of  20, 
(llTj)  which  will  give  -J  of  1  =  ^  of  the  dividend  as 
required.  In  reducing  this  expression  of  the  result,  say  4  in 
48,  12  times,  and  3  times  ^  are  -^,  equal  to  7|  barrels, 
which  is  the  answer  required. 

Another  Explanation. — First,  divide  by  20  as  if  it  were  20 
units,  which  gives  ^^;  but,  as  the  right  divisor  is  ^,  only 
I  of  20  units,  it  will  be  contained  3  times  as  often  as  20 
units  (151  j)  therefore,  multiply  that  quotient  by  3  to  ascertain 
how  many  times  ^^  is  contained  in  48,  which  gives  ^9t^'^^ 
==^=7-1-  barrels,  as  before. 

153*    Observation. 

Observe,  {132^)  that,  in  dividi?ig  by  a  fraction,  the  pro- 
cess consists  of  two  steps,  on  account  of  the  two  numbers  in  the 
divisor,  and  that,  either  step  may  be  taken  first,  provided 
the  reasoning  be  suited  to  the  process, 

1«I4.   Exercises  in  dividing  by  a  Fraction. 

In  like  manner,  solve  and  explain  the  following  problems, 
1.    To   how  many  poor   persons   could  9  dollars  be  dis- 
tributed, giving  them  |  of  a  dollar  apiece  ? 


FRACTIONS.  107 

:},  If  28  dollars  be  paid  for  If  Ions  of  hay,  what  is  the 
price  of  a  ton  ? 

3.  If  a  drunkard  drink  ^^  of  a  quart  of  rum  per  day,  how 
lor  g  would  9  quarts  last  him  ? 

L  If  a  moderate  drinker  drink  |  pint  of  brandy  per  day, 
ho  AT  long  would  8  pints  last  him  ? 

5.  How  long  would  2  barrels  of  flour  last  a  family  that 
consume  f  of  a  barrel  in  each  week? 

3.  If  28  bushels  be  sown  on  9J  acres,  how  much  is  that 
pe '  acre  ? 

7,  If  it  take  §  of  a  bushel  of  rye  to  sow  an  acre,  how 
rainy  acres  would  15  bushels  sow? 

3.  How  many  bottles  of  beer  holding  /-g-  of  a  gallon  each, 
could  be  filled  from  a  hogshead  holding  6S  gallons  ? 

9.  At  1^  dollars  a  bushel,  how  much  wheat  could  be 
bo  ight  for  20  dollars  ? 

10.  How  many  acres  would  it  take  to  produce  96  bushels, 
at  the  rate  of  15f  bushels  per  acre  ? 

11.  If  a  man  pay  21  dollars  for  pasturing  his  horse  16§ 
weeks,  how  much  is  that  per  week? 

12.  If  a  man  earn  6  dollars  in  f  of  a  month,  how  much  is 
theit  for  one  month  ? 

13.  In  what  time  can  a  man  build  28  rods  of  wall,  if  he 
build  f  j  of  a  rod  per  hour? 

14.  If  IJ  yards  of  cloth  be  put  into  a  coat,  how  many 
coats  may  be  made  from  30  yards  ? 

15.  At  I  of  a  dollar  a  bushel,  how  many  bushels  of  corn 
may  be  bought  for  125  dollars  ?    ^ 

16.  How  many  pairs  of  gloves  may  be  bought  for  12  dol- 
lars at  f  of  a  dollar  a  pair  ? 

17.  If  7|f  barrels  of  apples  be  bought  for  20  dollars, 
what  is  the  cost  of  one  barrel  ? 

18.  If  ll-j^j-  gallons  of  molasses  cost  3  dollars,  what  would 
be  the  cost  of  one  gallon  ? 

19.  Divide  128  by  ^^, 

20.  How  many  times  is  ^*-  contained  in  19  ? 

21.  How  many  times  is  f  contained  in  14? 

22.  How  many  times  is  ^  contained  in  9  ? 

23.  Divide  2  bv7|. 

24.  Whatpart*'of7is3? 

25.  What  part  of  7  is  f  ? 

26.  What  part  off  is  7? 


108  ARITHMETIC. 

27.  What  part  of  2^  is  2  ? 

28.  What  part  of  6 J  is  5  ? 

1S5.    Model  of  a  Recitation. 

1.  With  3f  dollars  how  many  yards  of  broadcloth,  at  9 
dollars  per  yard,  could  a  merchant  buy  ? —  How  many  yards 
of  cassimere,  at  2  dollars  per  yard  ?  —  How  many  yards  of 
satinet,  at  J  of  a  dollar  per  yard?  —  How  many  yards  of 
camlet  at  |-  of  a  dollar  per  yard?  —  How  many  yards  of 
velvet  at  2f  dollars  per  yard? 

He  could  buy  as  many  yards  of  broadcloth,  at  9  dollars 

■yi:§-  =  i  yards  of  broadcloth.  S  n  •'   ^ 

^  ^  -^  dollars    is    con- 

5^2-  ==  fi^  =  Hi  ya^ds  of  cassimere.        tained  times  in 

2 7-i^3  3^  9  =  41  yards  of  satinet.  ^^i^'  ^^^\^^^ 

9  ~*        -^  2  J  which  ascertain 

y-iisxb  =  ^^  =  ^  yards  of  camlet.  by  dividing  the 

P^-B  =  U=-Hi  yards  of  velvet.  «;-*-  °J  ^f^ 

He  could  buy  as  many  yards  of  cassimere,  at  2  dollars 
per  yard,  as  2  dollars  is  contained  times  in  3f  =  -^-  dollars ; 
which  ascertain  by  dividing  the  size  of  the  parts  by  2,  (II45) 
that  is,  making  the  parts  J  as  large. 

He  could  buy  as  many  yards  of  satinet,  at  |  of  a  dollar 
per  yard,  as  |  of  a  dollar  is  contained  times  in  ^-^-  of  a 
dollar ;  multiply  by  4,  by  making  the  parts  4  times  as  large, 
(IOO5)  to  ascertain  how  many  times  ^'  is  contained  in  ^^-; 
(Itil  •)  and,  since  |  is  3  times  as  much  as  ^,  and,  con- 
sequently, will  be  contained  only  -J-  as  often  as  ^,  divide  this 
quotient  by  3,  by  dividing  the  number  of  parts,  to  ascertain 
how  many  times  |  is  contained,  which  reduced  will  be  the 
answer  required. 

He  could  buy  as  many  yards  of  camlet,  at  f  of  a  dollar 
per  yard,  as  |  of  a  dollar  is  contained  times  in  3f  =  %7-  of  a 
dollar ;  multiply  by  8,  by  multiplying  the  size  of  the  parts,  to 
ascertain  how  many  times  |-  is  contained  in  -^/-,  and,  since  f 
will  be  contained  -J  as  often,  divide  this  quotient  by  5,  by 
dividing  the  size  of  the  parts,  to  ascertain  how  many  times  | 
is  contained,  which  reduced  will  be  the  answer  required. 

He  could  buy  as  many  yards  of  velvet,  at  2§=:|  of  a 
dollar  per  yard,  as  f  of  a  dollar  is  contained  times  in  3f  =  ^Z- 
of  a  dollar ;  multiply  the  number  of  parts  by  3  to  ascertain 
how  many  times  J  is  contained  in  ^,  and  divide  the  size  of 


FRACTIONS.  109 

the  parts  in  that  quotient  by  8  to  ascertain  how  many  times  f 
is  contained,  which  reduced  will  be  the  answer  required. 

Another  Explanation.  —  First,  divide  3f ,  or  %7_  by  8  as  if  it 
were  8.  units,  which  gives  -g^^;  but,  as  the  right  divisor  is  f, 
orly  ^  of  8  units,  (94)  it  will  be  contained  3  times  as  often 
as  8  units  ;  therefore,  multiply  that  quotient  by  3  to  ascertain 
he  w  many  times  f  is  contained ;  which  gives  %^-§f  =  f^  = 
^ii  yards  as  before. 

1«I6«   Observation. 

Observe,  that,  in  dividfng  a  fraction  hy  a  fraction^  the 
process  consists  of  two  steps,  either  ofiohich  may  be  taken  first ; 
thiit,  in  many  cases  there  are  two  loays  of  performing  each 
part  of  the  process,  on  account  of  the  two  numbers  in  the  divi- 
de.id  ;  but  that,  of  the  tiuo  ways,  that  is  to  be  adopted  which 
wiR  give  the  result  in  the  loiver  terms ;  that,  each  part  of  the 
process  is  to  be  expressed  and  explained  separately;  and 
fit'jolly,  that  the  process  is  to  be  performed  by  reducing  the 
expression  of  the  result  to  its  simplest  terms, 

1£>7«   Exercises  in  dividing  a  Fraction  by  a  Fraction. 

In  like  manner,  solve  an^  explain  the  following  problems, 

1.  At  f  of  a  dollar  a  bushel,  how  much  rye  may  be 
bought  for  5  of  a  dollar  ? 

2.  At  1^  of  a  dollar  a  bushel,  how  many  apples  may  be 
bought  for  |-  of  a  dollar  ? 

3.  How  many  bushels  of  turnips,  at  ^^  of  a  dollar  per 
bushel,  may  be  bought  for  J  of  a  dollar  ? 

4.  If  1  bushel  cost  :|  of  a  dollar,  how  many  apples  may  be 
bought  for  f  of  a  dollar  ? 

5.  At  J^  of  a  dollar  a  dozen,  how  many  dozen  of  lemons 
may  be  bought  for  If  dollars  ? 

6.  At  f  of  a  dollar  a  dozen,  how  many  oranges  may  be 
bought  for  5f  dollars  ? 

7.  At  ^  of .  a  dollar  a  pound,  how  many  figs  may  be 
bought  for  2J  dollars  ? 

8.  At  ^  of  a  dollar  a  bushel,  how  many  potatoes  may  be 
bought  for  4^  dollars  ? 

9.  At  f  of  a  dollar  a  bushel,  how  many  onions  may  be 
bought  for  \  of  a  dollar  ? 

10.  With  53  dollars,  how  many  pounds  of  butter,  at  j*^ 
of  a  dollar  a  pound,  may  be  bought  ? 

w 


10  AEITHMETIC. 

11.  If  f  of  a  pound  of  fur  is  sufficient  for  1  hat,  how 
many  hats  would  4^^  pounds  be  sufficient  for  ? 

12.  If  1    yard  of  linen  cost  f f  of  a  dollar,  how  much 
would  3|  dollars  buy  ? 

13.  If  If  yards  of  cloth  make  1  coat,  how  many  coats  may 
be  made  from  9j-  yards  ? 

14.  If  2^  bushels  of  oats  keep  1  horse  a  week,  how  many 
horses  will  1S|  bushels  keep  for  the  same  time  ? 

15.  If  2J-  bushels  of  oats  keep  a  horse  1  week,  how  long 
would  12|  bushels  keep  him  ? 

16.  Bought  3^  yards  of  cloth  ♦for  14|-  dollars;  what  did  I 
give  per  yard  ? 

17.  At  f  of  a  dollar  a  pound,  how  many  pounds  of  coffee 
may  be  bought  for  12J  dollars? 

IS.    If  4f  pounds  of  butter  serve  a  family  1  week,  how 
many  weeks  would  36|-  pounds  serve  them  ? 

19.  If  a  man  walk  a  mile  in  ^  of  an  hour,  how  far  would 
he  walk  in  5f  hours  ? 

20.  If  a  barrel  of  cider  last  a  cider-drinker  S^  months, 
how  many  barrels  would  he  drink  in  lOf  months  ? 

21.  If  the  stage  run  8y^  miles  per  hour,  how  long  would 
'  '*^  be  in  running  25^^  miles  ? 

22.  How  many  bushels  of  rye  at  f  of  a  dollar  per  bushel, 
may  be  bought  for  12/3-  dollars  ? 

23.  If  4j-  pounds  of  t^a  cost  32-^^  dollars,  what  is  that  per 
pound  ? 

24.  How  many  times  is  4^  contained  in  3^? 

25.  At  If  dollars  per  yard,  how  much  carpeting  may  be 
purchased  for  33^  dollars  ? 

26.  Divide  If  by  33f 

27.  Divide  33^  by  1^. 

28.  If  ^f  of  a  dollar  buy  a  pound  of  tea,  how  much  would 
3^  dollars  buy  ? 

29.  How  many  times  is  16§  contained  in  83^  ? 

30.  How  many  times  is  6^  contained  in  62^  ? 

31.  How  many  times  is  8 J  contained  in  66f  ? 

32.  How  many  times  is  18|  contained  in  37j  ? 

33.  How  many  times  is  4^  contained  in  33^  ? 

34.  At  J  of  a  dollar  a  bushel,  how  much  corn  can  be 
bought  for  ^  of  a  dollar  ? 

35.  At  3  dollars  a  yard,  how  much  velvet  may  be  bought 
for  I  of  a  dollar  ? 

36.  Divide^  by  3? 


FEACTIONS.  Ill 

37.  What  part  of  10  is  7  ? 

38.  What  part  of  3  is  |  ?    . 

39.  What  part  of  3  is  2^  ? 

40.  Divide  5^  by  10. 

41.  What  part  of  10  is  5^? 

42.  Divide  2f  by  7f . 

43.  What  part  of  7f  is  2f? 

44.  Divide  f  by  ^. 

45.  When  corn  is  |-  of  a  dollar  per  bushel,  what  part  of 
a  bushel  may  be  bought  for  f  of  a  dollar  ? 

46.  f  is  what  part  of  J  ? 

158.    Review   of   the    several   Ways   of   multiplying 
A  Fraction  by  a  Fraction. 

Multiply  -f^  by  f . 

(a.)  State  the  problem. 
bo,  1.  1^^  ==  -J.  (b.)  What  may  be  the  first  step  ? 

*    2.  ^^p^  =  yV  =  h  (c.)  Why  need  that  be  don6  ? 

3.  352x-5=T=A==-^-  W  How  may  that  be  done? 

4.  f^^  ==  20  =  |.   (e.)  Why  may  it  be  done  in  that 

way? 


(/.)  Result  of  the  first  step? 

**    ^-  -fsii  =  i'  (o-)  What  must  be  the  next  step? 

^i  6.  xV^^^  =  xt  =  i-  (^O  Why  must  that  be  done  ? 
"  7.  i^r4><5  =  A-  =  i-  ( ^- )  How  may  that  be  done  ? 
"    9-  xVSf  =  §§•=  -i-  U')  Why  may  it  be  done  in  that 

way? 
{k.)  Result  of  both  steps? 

1^0«   Model  of  a  Recitation. 

No.  1.  (a.)  The  product  should  be  ^  of  the  multiplicand, 
(144.)  (b.)  First  divide  by  5,  (c.)  to  obtain  (92)  h  W 
which  is  done  by  dividing  the  numerator,  (111)  by  5;  (e.) 
because  that  will  give  \  as  many  parts,  (/.)  or  -j^^'  (S-)  Next, 
multiply  by  4,  {h.)  to  obtain  |,  [i.)  which  is  done  by  dividing 
the  denominator  (100)  by  4 ;  [j.)  because  that  will  make  the 
parts  4  times  as  large,  {k,)  or  ^,  which  is  the  answer 
required. 

In  like  manner  explain  the  first  four  of  the  examples  ;  but 
explain  the  last  four  by  ttsing  the  numerator  of  the  multiplier 
in  the  first  step  of  the  process. 


112 


ARITHMETIC. 


160.   Review   of   the    several   Ways  of    Dividing   a 
Fraction   by   a   Fraction. 


Divide  Jf  by  f . 


No. 


^'  ifcisxT  ==  iS  =  f  • 
4.  X^^^ 


_    60    _ 

-TU"(y- 


(<z.)   State  the  problem. 
( 5.)  What  may  be  the  first  step  ? 
( c. )  Why  need  that  be  done  ? 
(d.)  How  may  that  be  done  ? 
(e.)  Why  may  it  be  done  in  that 
way? 
-  (/.)  Resuh  of  the  first  step. 
(g.)  What  must  be  the  next  step  ? 
(h.)  Why  must  that  be  done  ? 
{i.)  How  may  that  be  done  ? 
{j.)  Why  may  it  be  done  in  that 

way  ? 
(k.)  Result  of  both  steps. 


161.   Model  of  a  Recitation. 


No.  1.  (a.)  It  is  required  to  find  how  many  times  f  is 
contained  in  the  dividend,  ( 63 ) ;  (3.)  First,  multiply  by  5, 
(c.)  to  ascertain  how  many  times  ^  is  contained,  (OOj)  (d.) 
which  is  done  by  dividing  the  denominator  (106)  by  5; 
(e.)  because  that  will  make  the  parts  5  times  as  large,  (105) ; 
(/.)  or,  J^.  (g.)  Next,  divide  by  4,  (k.)  to  ascertain  how 
many  times  f  is  contained,  (i.)  which  is  done  by  dividing  the 
numerator,  (111)  by  4;  (j.)  because  that  will  give  J  as 
many  parts,  (k.)  or  f ,  which  is  the  answer  required. 

In  like  manner  ^  explain  the  first  four  of 'the  examples  ;  but 
explain  the  last  four  by  using  the  numerator  of  the  divisor  in 
the  first  step  of  the  process. 


DECIMAL   FRACTIONS.  113 


VII.    DECIMAL    FRACTIONS. 

I'SS.    Similarity-  of    Decimal    Fractions  to    Integral 
Numbers. 

In  integral  numbers  you  see  that  there  is  a  uniform  law, 
10  units  of  any  order  making  1  unit  of  the  jiext  higher  order, 
(1.0)  or  1  unit  of  any  order  making  10  units  of  the  next  lower 
Older;  that,  therefore,  the  units  of  the  different  orders  are 
written  together  in  places  appropriated  to  them,  according  to 
their  values;  and  that,  hence,  the  values  of  the  several  units 
aie  known  from  the  places  which  they  occupy. 

But  in  fractional  numbers,  you  see  that  there  is  no  such 
uniformity,  since  the  parts  may  be  of  any  size,  depending 
u )on  the  number  of  them  that  it  takes  to  make  a  unit;  that, 
therefore,  the  parts  of  different  sizes  cannot  be  written 
together  in  places  appropriated  to  them,  according  to  their 
values,  and  that,  hence,  the  values  of  the  parts  cannot  be 
k:iown  from  the  places  which  they  occupy ;  but  that  the  parts, 
whatever  may  be  their  size,  are  written  in  the  same  place,  at 
the  right  of  an  integral  number,  when  not  written  alone,  and 
always  accompanied  by  a  denominator  to  show  their  size, 

You  will  now  give  your  attention  to  a  kind  of  fractions  in 
which  there  prevails  the  same  uniformity  as  in  integral  num- 
bers; 10  parts  of  any  size  making  1  part  of  the  next  larger 
size ;  or  1  part  of  any  size  making  10  parts  of  the  next 
smaller  size,  (10) ;  and  therefore,  the  parts  of  different  sizes, 
are  written  together  in  places  appropriated  to  them,  according 
to  their  sizes ;  and  hence  the  different  sizes  of  the  parts  are 
known,  without  the  presence  of  their  denominator,  from  the 
places  which  the  parts  occupy ;  moreover,  all  the  operations 
of  addition,  subtraction,  multiplication,  and  division,  are  per- 
formed upon  them,  either  alone,  or  together  with  integral 
numbers,  precisely  as  upon  integral  numbers  alone;  care 
being  required  only  to  keep  the  point  of  separation  between 
the  integral  and  fraction^  parts  of  numbers, 
10^ 


m 


ARITHMETIC. 


*163.   Illustration   of   the   Local   Value    of    Decimal 

FiGCJRES. 

Observe,  m 
;  - 1-  =  f  ?>  =  m^  =  nn^  =  inU>  &c.  this  taWe,  that 
!  1    unit    is    re- 

■    ■  *  '  "tVj  =  tVtt^  =  TTHj^j  =  TyW(j'  &c.       duced  to  tenths, 
\  \  *  '       ^c,  ^  to  hun- 

'  '  ; TiTyj  =  T*^TTJ  =  Ti8wj  &c.       dredths,     &c., 

'     '   •  y^^       to       ^AOM- 

[   !  I  [ tttW>  =  TiTTjVorj  <^c.       sandths,      &c., 

.  .  .  ;  yx>aTy    to     ifew- 

;;;;'. TTTTyxru'  ^^'       thousandths, 

1.1111.  &€.,  by    mul- 

tiplying both 
terms  (  121  )  of  each  fraction  by  10,  and  by  10  again,  &:c. ; 
that  1  part  of  each  size  makes  10  parts  of  the  next  smaller 
size,  or  that  10  parts  of  each  size  make  1  part  of  the  next 
larger  size,  (10) ;  and  that  on  the  left,  1  part  of  each,  size  is 
arranged,  without  its  denominator,  according  to  the  values  of 
the  parts,  1  unit  being  written,  then  1  tenth  in  the  first 
place  at  the  right  hand  of  units,  1  hundredth  in  the  second 
place,  1  thousandth  in  the  third  place,  1  ten-thousandth  in  the 
fourth  place,  &c.  Any  other  digit  written  in  any  of  these 
places,  would  express  parts  of  the  size  for  which  that  place  is 
appropriated.  Hence,  the  values  of  these  parts,  or  any  num- 
ber of  parts  arranged  in  this  way,  according  to  their  values, 
may  be  known  without  their  denominator,  since  the  different 
parts  will  always  occupy  places  at  the  same  relative  distances 
from  the  unit's  place  ;  but  a  point  ( • )  must  be  prefixed  to  a 
fraction  to  distinguish  it  from  an  integral  number,  or  the 
integral  part  of  a  mixed  number. 

Such  fractions  are  called  Decimal  Fractions,  because  the 
parts  expressed  by  them  are  always  such,  that  it  takes  10  of 
them,  or  some  power  (40)  of  10  to  make  a  unit.  They 
differ  from  Common  Fractions,  only  in  the  uniformity  in  the 
values  of  the  parts  expressed  by  them,  and  consequently,  in 
the  manner  of  writing  them,  and  operating  by  them. 

164.   Mode  of  Reading  Decimal  Numbers. 

Since,  as  you  may  observe,  the  places  equidistant  from  the 
units,  on  each  side,  correspond  in  name,  except  that  the  ter- 


DECIMAL   fractions". 


116 


mination  of  the  fractional  names  is  ths^  the  manner  of  reading 
decimal  fractions  is  similar  to  that  of  reading  integral 
numbers. 

Observe^  also,  in  the  table,  that  1  =  10000  ten-thousandths, 
j-^^-=  1000  ten-thousandths,  xi-(j==  ^^^  ten-thousandth,  y^Vrr 
==10  ten-thousandth,  andY^^^=l  ten-thousandth;  con- 
sequently, the  whole  mixed  number,  1.1111,  may  be  read, 
eleven  thousand  one  hundred  and  eleven  ten-thousandths, 
piecisely  the  same  as  an  integral  number,  except  at  last, 
speaking  the  denominator  of  the  last  figure,  which  is  also  the 
C('mmon  denominator  of  this  and  the  other  figures  in  the 
number,  as  may  be  observed  in  the  table.  But  the  better 
way  is  to  read  the  integral  and  fractional  parts  separately. 
Thus :  One,  and  one  thousand  one  hundred  and  eleven 
ten-thousandths. 

The  denominator  of  the  last  figure,  or  the  common  denomi- 
nator of  all  the  figures  in  the  numerator,  may  be  known  from 
tie  fact  that  it  will  always  consist  of  one  more  figure  than 
the  decimal  places  occupied  by  the  numerator,  or  1  with  as 
many  ciphers  as  the  numerator  occupies  decimal  places. 

165«   Exercises  in  Reading  Decimal  Numbers. 
In  like  manner,  read  the  following  numbers. 


L 
2. 
3. 

4. 
5. 
6. 

7. 

8. 

9. 

10. 


5.111. 
3.12. 
2.6. 
.2. 
.25. 
.75. 
.125. 
17.3. 
144.16. 
3456.4. 


11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


252.5. 
25.25. 
2.525. 
.2525. 
40.5. 
4.05. 
.405. 
306.1. 
30.61. 
3.061. 


21. 

.05. 

31. 

2.40003. 

22. 

.005. 

32. 

2.400305. 

23. 

.0005. 

33. 

.5. 

24. 

.00005. 

34. 

.50. 

25. 

.000005 

35. 

.500. 

26. 

.007. 

36. 

.5000. 

27. 

.00007. 

37. 

.050. 

28. 

.072. 

38. 

8.0900. 

29. 

3.0407. 

39. 

.0000009. 

30. 

cim. 

3.4007. 

iL  NUMBE 

40. 

rs. 

1.00000080 

106.    Mode  of  writing  Decimal 

In  writing  a  decimal  fraction,  it  should  be  made  to  occupy 
as  many  places  as  it  requires  ciphers  in  its  denominator. 
Therefore,  following  the  point,  write  as  many  ciphers  as  the 
number  of  decimal  places  required  exceeds  the  number  of 
figures  expressing  the  numerator ;  then*  write  such  figures 
as  will  express  the  numerator;  and  the  fraction  will  be 
properly  expressed.         ^ 


116 


A.RITHMETIC. 


107.   Model  of  a  Recitation. 

Write  twelve,  and  one  thousand  and  sixteen  ten-niil- 
lionths. 

First,  write  twelve  and  the  point,  thus ;  12. ;  then,  as  the 
decimal  must  occupy  seven  places,  and  it  requires  only  four 
figures  to  express  the  numerator,  write  three  ciphers,  and 
one  thousand  and  sixteen,  thus;  12.0001016,  which  is  the 
number  required. 

168.   Exercises  in  writing  Decimal  Numbers. 

In  like  rrunvner^  write  the  following  numhen^  expressing 
the  fractions  decimally. 

16.    tVttV 

18.    Tf«(T. 

21.  Seventeen,  and  four  hundred  and  nine  thousandths. 

22.  Six,  and  sixty-five  thousandths. 

23.  Seven,  and  seven  ten-thousandths. 

24.  Ten  thousand  eight  hundred  and  nine  hundred-thou- 
sandths. 

25.  Twenty-six,  and  fifteen  millionths. 
Three,  and  one  hundred  and  one  ten-thousandths. 
Four,  and  twenty-five  hundred-thousandths. 
Eight,  and  six  hundred  and  four  millionths. 
One,  and  sixty  thousand  and  five  ten-millionths. 
Two,  and  thirty  thousand  hundred- thousandths. 
How  many  thousandths  in  .2? 
How  many  hundredths  in  2.5  ? 
Reduce  -fjy  to  thousandths. 
Reduce  xV^^V  to  its  lowest  terms. 
Reduce  .25  to  its  lowest  terms  in  a  common  fraction. 
Reduce  .3125  to  its  lowest  terms. 
Reduce  ^,  y^,  and  y^^j  to  thousandths  and  add 


1. 

27^^. 

6. 

I^xtjVtt. 

11. 

ISx^TT. 

2. 

14x5x7. 

7. 

A- 

12. 

It**!.. 

3. 

108xV. 

8. 

T^xy. 

13. 

17x||^. 

4. 

73x«Tr. 

9. 

TTrxrxr. 

14. 

TTTTTO'TJ'. 

5. 

^100* 

10. 

nj-fuu". 

15. 

^- 

26. 

27. 

28. 

29. 

30. 

31. 

32. 

33. 

84. 

35. 

36. 

37. 
them. 

38. 
and  add  them. 


Reduce  -fj^ ,  xttWj  ^^^  x^j  to  a  common  denominator, 


DECIMAL   FRACTIONS.  117 

1€>9«  Federal  Money  expressed  by  Decbial  Numbers. 

-federal  money  is  the  metallic  money  which  is  coined  by 
the  authority  of  the  United  States.  It  consists  of  eagles, 
dollars,  dimes,  cents,  and  mills ;  the  values  of  which,  as  you 
may  observe  in  the  folio v/ing  table,  correspond  to  decimal 
nu  nbers ;  1  coin  of  either  denomination  equaling  10  of  the 
next  lower;  or  10  coins  of  either  denomination  equaling  1  of 
the  next  higher,  ( 10.)  But  the  mill  is  only  an  imaginary  coin. 
n  commerce,  eagles  are  expressed  in  dollars,  and  dimes  in 
ceiits.  The  dollar  is  considered  the  unit,  and  cents  and 
mi  Is,  decimal  fractions  of  a  dollar.  Hence,  numbers  express- 
ing- Federal  money  are  precisely  like  numbers  in  decimal 
fra:tions,  and  they  are  made  to  express  Federal  money  by 
prefixing  to  them  this  character  ($.) 

2  ri  •  .  J      2  J 

^  Q  p  o  S  —  '^^  Read, 

1  =  10=  100  =  1000  =  10000  =  $10. ...  Ten  dollars. 

1  =    10  =    100  =   1000  =  $  1. .  . .  One  dollar. 

1=      10=     100  =  $     .10.  Ten  cents. 

1=        10  =  $     .01.  One  cent.  , 

1  =  $     .001  One  mill. 


11111  (fljil    111     5  Eleven  dollars,  eleven 

lllli  ^11.111    ^  cents  and  one  miU. 

170*    Reduction  of  Federal  Money  illustrated. 

1.  In   25  how  many  hundredths?  —  how   many   thous- 
andths ? 

Since  there  are  ^%%  in  1,  there 
2500  hijpdredths.  will  be  100  times  as  many  hun- 

dredths as  units ;  therefore,  multi- 
25000  thousandths.         ply  the  units  by  100,  by  annexing 

two  ciphers,  (3^1:.) 
There  will  be  1000  times  as  many  thousandths  as  units; 
therefore,  annex  three   ciphers  to  25,  which  will  give  the 
answer  required. 

2.  In  $25,  how  many  cents  ?  —  how  many  mills  ? 

Since  there  are  100  cents  in  $1,  there 
2500  cents.         will  be  100  times  as  many  cents  as  dol- 
lars;  therefore,  annex  two  ciphers,  (10) 
25000  mills.         to  25,  making  2500  cents,  which  is  the 
answer  required. 


lis  ARITHMETIC. 

There  will  le  1000  times  as  many  mills  as  dollars;  there- 
fore, annex  three  ciphers  to  the  dollars,  which  will  reduce 
the  dollars  to  mills,  as  required. 

Observe  that^  hij  'pointing  off  the  ciphers  annexed  in  these 
examples^  that  is,  putting  a  point  between  the  25  and  the 
ciphers,  the  hundredths  and  thousandths  will  be  reduced  to 
units  again,  and  the  cents  and  mills  to  dollars  again, 

171.   Model  of  a  Recitation. 

1.    In  25125  mills  how  many  cents  ?  —  how  many  dollars  ^ 

Since  in  1  cent  there  are  10  mills, 

2512.5  cents.         there  will  be  y^,  or  .1  as  many  cents  as 

mills  ;  therefore,  divide  by  10,  by  point- 

$25,125  ing  off  one   figure  at  the  right  hand, 

( 10) ;  for  thus  the  tens  become  units, 

and  the  other  figures,  also,  are  all  brought  one  degree  lower 

There  will  be  tx^xj-,  or  .001  as  many  dollars  as  mills , 

therefore,  point  off  three  figures,  (10) ;  for  thus  the  thousands 

become  units,  and  all  the  other  figures  are  also  brought  three 

degrees  lower. 

173.   Exercises  in  the  Reduction  of  Federal  Money. 

In  like  manner,  solve  and  explain  the  following  'problems, 

1.  In$16,  how  many  cents? 

2.  How  many  mills  in  $16  ? 

3.  In  12000  mills  how  many  cents  ? 

4.  How  many  dollars  in  12000  mills  ? 

5.  In  75  cents  how  many  mills  ? 

6.  In  $8.25  how  many  cents  ? 

7.  In  $5,125,  how  many  mills  ? 

8.  How  many  cents  in  $5,125?  ♦ 

9.  In  $3,375,  how  many  dollars,  cents  and  mills  ? 

10.  In  16125  mills  how  many  dollars? 

11.  In  12548  cents,  how  many  dollars  ? 

12.  Reduce  $37.50  to  cents. 

13.  Reduce  75625  mills  to  dollars  ? 

14.  Reduce  984  mills  to  dollars. 

15.  Reduce  $.75  to  cents. 

16.  Reduce  $.125  to  mills. 

17.  Howmany  mills  in  $1.25? 

18.  How  many  cents  in  $2,375? 

19.  In  12345  mills  how  many  dolUrs,  cents  and  mills  ^ 

20.  How  many  times  10  in  85  ? 


DECIMAL   FRACTIONS.  119 

21.  Divide  625  by  100. 

22.  Divide  1836  by  1000. 

23.  What  is  y^^  of  1728? 

24.  How  many  times  100  in  1276  ? 

25.  Reduce  12.25  to  hundredths. 

178.   Model  of  a  Recitation. 

1.  Bought  1  barrel  of  flour  for  $6.75,  10  pounds  of  cof- 
fee for  $2.20,  7  pounds  of  sugar  for  $.875,  12  pounds  of 
bjtter  for  $2,  1  pound  of  raisins  for  $.125,  and  2  oranges  for 
$.06.     What  was  the  whole  amount  ? 

Arrange  the  numbers  together  so  that  the  figures  of  each 
denomination,  may  stand  in  a  column  by  them- 
6.75  selves,  and   proceed   as   in   the   addition   of 

2.20  integral  numbers,  (20.) 

.875  The  10  mills  of  the  first  column   make  1 

2.  cent,  (I6O5)  which  added  with  the  first  column 

.125         of  cents  make  21  cents,  equal  to  1  cent,  which 

.06  write,  and  2  dimes ;  which  add  with  the  other 

dimes,  making  20  dimes,  equal  to  2  dollars  ; 

$12.01  write  a  cipher  in  the  dimes'  place,  or  second 

place  of  cents,  and  add  the  2  dollars  'with  the 
other  dollars,  making  12  dollars,  which  written  at  the  left  of 
the  point,  make  $12.01,  the  answer  required. 

2.  Mr.  Farmer  having  a  pasture  of  25  acres,  fenced  off 
2.375  acres  to  plant  with  potatoes ;  hoV  many  acres  remained 
in  the  pasture? 

Write  the   subtrahend   under   the   minuend,  placing   the 

figures  of  each  denomination  under  those 

25.  of  the  same  denomination,  and  proceed  as 

2.375  in  the  subtraction  of  integral  numbers, 

{33^.)    Since  there  are  no  thousandths 

22.625  acres.  from  which  to  take  the  5,  reduce  1  of 
the  5  units  to  tenths,  (IO5)  making  10, 
one  of  which  (leaving  9,)  i:educe  to  hundredths,  making  10, 
one  of  which  (leaving  9,)  reduce  to  thousandths,  making  10, 
from  which  subtract  the  5,  and  5  thousandths  remain,  which 
write ;  7  hundredths  from  9  hundreths  leave  2  hundredths, 
which  write ;  3  tenths  from  9  tenths  leave  6  tenths,  which 
write; 2  units  from  4  units  leave  2  units,  which  write;  and 
blank  from  2  tens  leaves  2  tens,  which  t^rite;  making  22.625 
acres,  which  is  the  answer  required. 


3^ 


120 


ARITHMETIC. 


174.    Exercises   in   adding  and    subtracting    Decimal 
Numbers. 

In  like  manner^  solve  and  explain  the  following  problems y 
taking  care  to  keep  a  point  between  the  integral  and  frac- 
tional parts  of  every  number. 

1.  Bought  a  pair  of  oxen  for  $76.50,  a  horse  for  $75,  and 
a  cow  for  $25.75;  what  was  the  whole  amount? 

2.  A  man  gave  $4.75  for  a  pair  of  boots,  and  $2.25  for  a 
pair  of  shoes ;  how  much  more  did  the  boots  cost  than  the 
shoes  ? 

3.  A  man  bought  a  cow  and  calf  for  $28,375,  and  sold  the 
calf  for  $3,625,  what  did  the  cow  cost  him  ? 

4.  Bought  a  horse  for  $92,  but  sold  him  so  as  to  lose 
$15.25 ;  for  how  much  was  he  sold  ? 

5.  What  is  the  whole  cost  of  a  cart  at  $17,625,  a  wagon  at 
*S.50,  a  plough  at  $7,333,  a  rake  at  $.42,  a  hoe  at  $.60, 

and  a  pitchfork  at  $.875  ? 

6.  How  much  cloth  in  6  pieces  measuring  as  follows, 
25.5  yards,  27.75  yards,  28.125  yards,  30  yards,  29.375 
yards,  and  26.5  yards  ? 

7.  A  merchant  having  a  piece  of  cloth  measuring  25 
yards,  sold  from  it  1.875  yards  for  a  coat,  and  1.125  yards 
for  a  pair  of  pantaloons ;  how  much  was  there  left  in  the 
piece  ?  ' 

8.  Mr.  Farmer  took  to  market  32  bushels  of  potatoes  in 
one  load,  and  peddled  them  as  follows  :  5.3  bushels  for  $2.75, 
4.25   bushels   for   $2,125,   6.75   bushels   for   $3,375,    10.5^ 
bushels  for  $5.25,  and  the  rest  of  the  load  for  $2.50;  how  ^ 
much  did  he  sell  at  the  last  sale ;  and  how  much  did  he  get 
for  his  load  ? 

9.  A  man  owing  $253,  paid  $187,375,  how  much  did  he 
then  owe  ? 

10.  Add  together  10.0625,  5.1875,  %.5,  and  4.25. 

11.  How  much  is  15.5  +  2.75  +  3.75— 12  ? 

12.  What  is  the  sum  of  192.423  and  20.58? 

13.  What  is  the  difference  between  12.5  and  6.25  ? 

14.  ^\hat  is  the  sum  and  difference  of  245.0075  and 
234.9925  ? 

15.  Subtract  2yV  ^^om  4yVu 

17^.   Model  of  a  Recitation. 

I.  If  1.75  yards  be  required  for  1  coat,  how  much  would 
be  required  for  7  coats  ? 


DECIMAL    FRACTIONS.  121 

1  175  Arrange  the  multiplicand  and   multi- 

^  plier,  and  proceed  as  in  the  multiplication 

of  integral  numbers,  (S7)  and  the  product 

12  25  vards  would  be  1225;   but,  since  7  times  175 

^  *  things  of  any  kind  will  be  1225  things 
of  the  same  kind,  7  times  175  hundredths  will  be  1225  hun- 
dredths, or  12.25. 

But,  to  analyze  it,  say  :  7  times  5  hundredths  are  35  hun- 
dradths,  equal  to  5  hundredths,  which  write  in  the  place  of 
hindredths,  and  3  tenths,  (lOj)  which  add  with  7  times  7 
te  iths,  making  52  tenths,  equal  to  2  tenths,  which  write  in 
th3  place  of  tenths,  and  5  units,  which  add  with  7  times  1 
ur.it,  making  12  units,  which  write  at  the  left  of  the  point, 
ard  the  result  will  be  the  answer  required. 

2.  At  $175  per  acre,  what  would  be  the  cost  of  .7  acres, 
or,  more  properly,  .7  of  an  acre  ? 

Since  1  acre  costs  $175,  .7  of  an  acre  would 

?'175  ^^^^  *'''  ^^^^^  ^^  much,  or  .7  as  much.     First, 

ry  multiply  by  7,  as  if  it  were  7  units,  which  gives 

•  $1225.     But,  the  right  multiplier  being  .7,  only 

(t.-j 22  5  ro  of  ^  units,  the  right  product  should  be  only  -^ 

of  1225  ;   therefore,  divide  this  product  by  10, 

by  removing  the  point  one  place  farther  to  the 

left,  (IO5)  which  gives  $122.50,  the  answer  required. 

3.  What  would  be  the  price  of  2.25  cords  of  wood,  at 
$5,375  per  cord  ? 

Since  1  cord  costs  $5,375,  2.25  cords 

$5,375  would  cost  2.25  times  as  much.     225 

2.25  times  5.375  would  be  1209.375.     But, 

~26875  ^^^  multiplier  being  only  y^^  of  225,  the 

107^50  product  will  be  only  y^  of  1209.375; 

10750  therefore,  divide  by  100,  by  pointing  off 

two  more  figures   (171)   for  decimals, 

$12.09375  making  $12.09375,  which  is  the  answer 

required. 

4.  Multiply  .125  by  .03. 

3  times  .125  would  be  .375.     But  the  multi- 

.125         plier  being  ^-^^  of  3,  the  product  will  be  ^-^^ 

.03         of  .375  ;  therefore,  divide  by  100,  by  removing 

the  point  (163)  two  place§  farther  to  the  left. 

.00375         But  you  must  make  those  places  in  this  exam- 
ple, by  prefixing  ciphers. 
11 


4 


122  arithmetic. 

170*   Proof  of  the  Pointing  in  the  Multiplication  of 

Decimals. 

Observe,  ihat^  in  the  preceding  examples,  {V75^)  each 
product  has  as  many  decimal  figures  as  all  its  fg,ctors.  This 
will  hold  true  in  all  cases  ;  and  this  tfuth  may  he  applied  to 
prove  the  pointing  of  the  product ;  for,  if  the  factors  be  con- 
sidered as  integral  numbers,  the  product  would  be  integral ; 
and,  since  for  every  removal  of  the  point  one  place  to  the  left, 
in  either  factor,  that  factor  becomes  y\j-  as  large,  (lO,)  and, 
consequently,  the  product  also  becomes  yV  ^^  large,  the  pro- 
duct must  be  divided  by  10,  [which  is  done  by  removing  the 
point  (10)  on£  place  to  the  left,)  for  every  decimal  figure 

IN  ALL  THE  FACTORS. 

lyy.   Exercises  in  the  Multiplication  of  Decimal  Num- 
bers. 
In  like  manner,  solve  and  explain  the  following  problems, 

1.  How  many  yards  of  cloth  would  be  refjuired  for  5  pairs 
of  pantaloons,  if  1.25  yards  be  put  into  each  pair  ? 

2.  What  cost  8  yards  of  cloth,  at  S2.875  per  yard  ? 

3.  How  many  dollars  in  8  ninepences,  if  $.125  make  1 
ninepence  ? 

4.  If  $.0625  make  1  fourpence-halfpenny,  how  much  ia 
16  fourpence-halfpennies  ? 

5.  How  much  would  a  man  receive  for  5  barrels  of  pork 
at  $17.25  per  barrel  ? 

6.  At  $5.50  per  yard,  what  cost  10  yards  of  broad  cloth  ? 

7.  At  $.05  per  pound,  what  cost  100  pounds  of  rice  ? 

8.  At  $.20  per  pound,  what  cost  1000  pounds  of  butter  ? 

9.  What  cost  60  pounds  of  candles,  at  $.17  per  pound  ? 

10.  What  cost  12  dozen  of  eggs,  at  $.125  per  dozen  ? 

11.  Multiply  5.333  by  8. 

12.  Multiply  .464  by  25. 

13.  How  much  is  50  times  .05  ? 

14.  What  is  the  amount  of  the  following  bill  ? 

Mr.  John  Debtor,  1^°^^"'  "^""^^  ^'  ^^^- 

Bought  of  Charles  Creditor, 
7  yds.  Broad  Cloth,  «®  $5.50    per  yard, 

5    "     Cassimere,  (a>     1.50      "      " 

12    *'     Striped  Jean,  (cb      .375    "       " 

15    "     Bleached  Sheeting,  (®      A^      "       '' 
27    ''     Brown  "         (t      .125    ''      " 


DECIMAL   FRACTIONS.  123 

15.  If  a  barrel  of  flour  cost  $6,  what  cost  .5  barrels  ? 

16  At  $25  a  ton,  what  cost  .7  of  a  ton  of  hay  ? 

17.  At  $6  per  yard,  what  cost  .25  of  a  yard  of  cloth  ? 

18.  At  $8  per  cord,  what  cost  .75  of  a  cord  of  wood  ? 

19.  At  $5D  per  acre,  what  cost  .125  of  an  acre  of  land  ? 

20.  What  is  the  amount  of  the  following  hM  ? 

Mr.  Jacob  Shem,  ^^l^-"'  ^""^  ^'  ^^^^- 

Bought  of  Israel  Ham, 
JJ7.5      yards  German  Broad  Cloth,  <»  $9  per  yd. 
tl5.75       "      French        "         "      o  $7    "     " 

18.125     "      English     Cassimere,   ®  $3    "     " 
IM.375     "      American         "  ®  $2    "     " 


21.  Multiply  144  by  .5. 

22.  What  is  .5  of  1728? 

23.  Multiply  512  by  .25. 

24.  What  is  .75  of  856? 

25.  Multiply  1840  by  .125. 

26.  What  is  .625  of  1000  ? 

27.  Multiply  75  by  .004. 

28.  What  is  .0003  of  3000? 

29.  At  $.96  a  gallon,  what  costs  .4  gallons  of  oil  ? 

30.  At  $.50  a  yard,  what  costs  .5  yards  of  cloth  ? 

31.  Multiply  .3  by  .6. 

32.  What  is  .4  of  .7  ? 

33.  Multiply  .25  by  .5. 

34.  What  cost  1.5  yards,  at  $.12  per  yard  ? 

35.  What  cost  4.12  yards,  at  $.50  per  yard  ? 

36.  What  is  the  amount  of  the  following  bill  ? 

Mr.  Reuben  Retail,  ^°^'°"'  ^""^  ^'  ^^^' 

Bought  of  Warren  Wholesale, 
6.5    dozen  Spelling-Books,      (®  $  1.75    per  doz. 
8.25      "      Young-Readers,     (®  $  2.875   "      " 
10.75      "      National-Readers,  (8>  $  7.50     "      " 
9.25      "      Testaments,  (Q  $  3.625   "      " 

4.5       "      Polyglot  Bibles,     (©$10.50     "      « 


124  ARITHMETIC. 

37.  What  is  the  product  of  .204  multiplied  by  1.4  ? 

38.  How  much  is  11.03  times  .1109  ? 

39.  Multiply  .04  by  .004. 

40.  What  is  .0006  of  .0012  ? 

41.  Multiply  1.006  by  .002. 

42.  Multipl||>  .062  by  .003. 

43.  How  much  is  .0004  of  .025  ? 

44.  What  is  the  second  power  (49)  of  .5  ? 

45.  What  is  the  second  power  of  .25  ? 

46.  What  is  the  third  power  of  .5  ? 

47.  What  is  the  fourth  power  of  .5  ? 

178.   Model  of  a  Recitation. 

1.  If  4  books  cost  $3,  how  much  would  that  be  apiece  ? 

One  book  being  J  of  4  books,  the  price 

4)  3  0  rS  75        ^^  ^  ^^^^  should  be  J  of  the  price  of  4 

2  8      *  books.    ^  of  3  dollars  would  be  |  of  a  dol- 

lar,  (94:5 )  ^^  ^  common  fraction  ;  but  the 

2Q  answer    may  be    obtained    in   a   decimal 

2Q  form.     Thus,  J  of  3  dollars  not  being  a 

whole  dollar,  reduce  the  3  dollars  to  dimes, 

or  tenths,  (ITOj)  making  30  tenths,  J  of 
which  is  .7,  and  2  dimes,  or  tenths,  re- 
maining, which  reduce  to  cents,  or  hundredths,  making  20 
hundredths,  J  of  which  is  .05,  which,  written  with  the  .7, 
makes  $.75,  the  answer  required. 

2.  A  man,  having  3  acres  of  land,  divided  it  into  8  equal 
house-lots.     How  many  acres  in  each  lot  ? 

g  Each  lot  would   contain  J  of  3 

Q\  Q  n  /  Q-r^  ««,.^o  acres,  which  is  §  of  an  acre  ;   but 

o)  J.U  (.J7o  acres.         ^i-  r     .•  -l        j       j 

OA  this  common  traction  may  be  reduced 

to  a  decimal  fraction.     Thus,  J  of  3 

QQ  acres  not  being  a  whole  acre,  reduce 

en  the  3  acres  to  tenths,  making  (163) 

30  tenths,  J  of  which  is  .3,  and  .6 

^Q  remaining,  which    reduce    to    hun- 

Ar.  dredths,  making   60   hundredths,  ^ 

of  which  is  .07,  which  write,  and  .04 

remaining,  which  reduce  to  thou- 
sandths, making  40  thousandths,  J  of  which  is  .005,  which 
write,  making  .375  of  an  acre,  the  answer  required. 


DECIMAL   FRACTIONS.  125 

3.   At  $3,375  for  9  gallons  of  molasses,  what  would  be  the 
cost  of  1  gallon  ? 

One  gallon  would  cost  -J  of  the  price  of  9 
(,v  o  rt^c        gallons.     ^  of  3  dollars  not  being  a  whole  dol- 

*^  '  _J lar,  reduce  the  3  to  tenth s^  making,  with  the  3 

<^  rt^^  tenths,  33  tenths,  ^  of  which  is  .3,  and  .6  re- 
maining,  which  reduce  to  hundredths,  making, 
with  the  7  hundredths,  67  hundredths,  \  of 
wiich  is  .07,  which  write,  and  .04  remaining,  which  reduce 
to  thousandths,  making,  with  the  5  thousandths,  45  thou- 
sandths, \  of  which  is  .005,  which  write,  making  $.375,  which 
is  the  answer  required. 

ITO.   Exercises    in    reducing    Common    Fractions    to 
Decimal  Fractions. 

In  like  manner^  solve  and  explain  the  following  problems. 

1.  If   12.25  yards   be   required  for  7  coats,  how  much 
W3uld  be  required  for  1  coat  ? 

2.  If  $.375  be  paid  for  3  boy's  tickets  for  admission  to  a 
concert,  what  would  be  the  price  of  1  ticket  ? 

3.  If  4  rides  in  a  car  cost  $6,  what  is  the  cost  of  1  ride  ? 

4.  If  3  bushels  of  apples  be  divided  among  4  men,  what 
would  be  each  man's  share  ? 

5.  If  12  dozen  of  eggs  cost  $1.50,  how  much  is  that  per 
dozen  ? 

6.  If  3  acres  of  land  be  fenced  off  into  5  equal  parts,  how 
many  acres  in  each  part  ? 

7.  Reduce  f  to  a  decimal  fraction. 

8.  Reduce  |  to  a  decimal  fraction. 

9.  Reduce  |  to  a  decimal  fraction. 

10.  Reduce  ^  to  a  decimal  fraction. 

11.  Reduce  -f^  to  a  decimal  fraction. 

12.  How  many  times  is  24  contained  in  6  ? 

13.  How  many  times  is  12  contained  in  28.8  ? 

14.  Divide  17.28  by  48. 

15.  Divide  1.44  by  72. 

16.  Divide  4.096  by  64. 

17.  Reduce  ^VV  ^^  ^  decimal  fraction. 

18.  How  many  times  is  50  contained  in  2.5  ? 

19.  How  many  times  is  100  contained  in  5  ? 

20.  Divide  3.75  by  8. 

21.  Divide  2.5  by  4. 

11# 


126  ARITHMETIC. 


180.  Model  of  a  Eecitation. 

1.  Divide  54.32  by  40. 

The  factors  10  and  4  composing  40,  first 
A)  5  4*^2         divide  by  10,  by  removing  the  point  (10)  one 

_^ place  farther  to  the  left,  to  obtain  -^jj  of  the 

1  rjro  dividend,  which  divide  (118)  by  4,  to  obtain 
4  of  tVj  0^  A-  of  the  dividend,  which  will  be 
the  answer  required. 

2.  How  many  times  is  35000  contained  in  31.5  ? 

The    factors    1000,  7,   and    5,   composing 
7^  A01C         35000,  first  divide  by  1000,  by  removing  the 

^ '_ point  (10)  three   places  farther  towards  the 

5^  004-^  ^^^'  ^^   obtain  iwau  of  the    dividend,  which 

^  '_ '_        divide   by   7,  to    obtain  ^   of  xxrW'  or  ytjVd 

/^Q/^Q  (118)  of  the  dividend,  which  divide  by  5,  to 
obtain  -^  of  tuVtu  or  -g^^iyxr  of  the  dividend, 
which  will  be  the  answer  required. 

181.  Observation. 

Observe  that,  when  the  divisor  is  a  certain  number  of 
tens,  hundreds,  or  thousands,  ^c,  it  is  more  convenieiit  to 
divide  first  by  one  ten,  hundred,  or  thousand,  <^c.;  then  divide 
that  quotient  by  the  other  factor  of  the  divisor. 

182.  Exercises   in   dividing   by   Units    of   the    higher 

Orders. 

In  like  maimer,  solve  and  explain  the  following  problems, 

1.  How  many  times  is  500  contained  in  1775  \ 

2.  Divide  6.25  by  250. 

3.  How  many  times  9000  in  63459  ? 

4.  What  is  the  quotient  of  129.6  divided  by  1200  ? 

5.  Divide  3.651  by  30. 

6.  How  many  tons,  of  2000  pounds  each,  in  16948.25 
pounds  ? 

7.  What  part  of  an  hour  is  21  minutes  ? 

8.  How  many  years  would  it  take  a  man  to  save  $5750, 
at  $500  per  year  ? 

9.  How  many  months,  at  $60  per  month,  would  it  take  a 
man  to  earn  $1296  ? 

10.  Divide  45.6855  by  1500. 


DECIMAL    FRACTIONS.  127 

11.  Divide  8943.75  by  75000. 

12.  What  part  of  a  ton,  or  2000  pounds,  are  1728  pounds  ? 

13.  How  many  times  is  42000  contained  in  586.488  ? 

14.  Reduce  14784  minutes  to  hours. 

15.  How  many  eagles,  of  $10  each,  in  $362.50  I 

1  §3.   Illustration  of  Infinite  Decimals. 

1.    If  $1  be  paid  for  9  writing-books,  how  much  would 
that  be  apiece  ? 

If    9    books    cost    $1,  one    book 

■j  would  cost  ^  of  a  dollar.     But  to  re- 

9^10C^111-U         duce^  to  a  decimal  form,  (ITS)  annex 

Q  •  a  point  and  a  cipher  to  the  numera- 

tor,  and  divide  it  by  the  denominator, 

1Q  which  will  give  .1  for  the  first  quo- 

Q  tient  figure.      A  cipher  annexed  to 

the  remainder,  gives  .01  for  the  quo- 

1^  tient ;   a  cipher  anilexed  to  this  i*- 

q  mainder,  gives  .001  for  the  quotient; 

and  thus,  continuing  without  limit, 

1  the  same  remainder  would  recur,  and 

the  same  figure  would  be  repeated  in 
the  quotient.     This  quotient,  and  the 
like,  are  called  Infinite  Bechnals. 
When  a  quotient  figure  thus  repeats,  it  is  called  a  repe- 
tend;  and  the   fact  of  its  being  a  repetend,  is   denoted   by 
placing  a  point  over  the  first  figure,  omitting  the  rest.     Thus, 
.i  =  .lll&c.==|;  .2  =  .222  &c.  =  I ;  .5  =  .55  &c.  =  |; 
and  any  figure,  thus  repeating,  is  so  many  times  -J.     There- 
fore, to  reduce  any  repetend  of  one  repeating  figure  to  a  com- 
mon fraction,  you  need  only  make  the  repeating  figure  the 
numerator,  and  9  the  denominator,  and  the  result  will  be  a 
fraction  of  1  in  the  next  higher  place. 
2.    Reduce  -^^  to  a  decimal. 

_1^  .  .  When  two  or  more  figures 

99)1.00(.0101  &c.  =  .01         repeat,  as  in  this  example,  the 

99  repetend  is  denoted  by  a  point 

—  over  both  the  first  and  last  of 

100  the  repeating  figures. 

Since    .61  =  ^^,   any    two 
figures  thus  repeating,  equal  so  many  thnes  ^^j  ^^^  i"^  ^i^^^ 


128  ARITHMETIC. 

manner,  three  repeating  figures  equal  so  many  times  ^^, 
&c. 

Therefore,  to  reduce  any  repetend  to  a  common  fractimi^ 
take  the  repeating  figures  for  a  numerator^  and  as  many  95, 
for  a  denominator,  and  the  result  loill  be  an  equal  fraction 
of  1  in  the  next  higher  place, 

184.  Model  of  a  Recitation. 

1.  Reduce  ^  to  a  decimal  fraction,  and  back  again  to  a 
common  fraction. 

i  =  .16  =  3V  +  fofTV=iV  +  A  =  /Tr  +  7V  =  i*  =  i- 

2.  Reduce  -^^  to  a  decimal,  and  back  again. 

^    sfA===.2Mf=A  +  if  of3-V==//^  +  ^f^===37^. 

185.  Exercises  in  the  Reduction  of  Infinite  Decimals. 

In  like  manner^  solve  and  explain  the  following  problems, 

1.  Reduce  -j^  to  a  decimal,  and  back  again  to  a  common 
fraction. 

2.  Reduce  ^  to  a  decimal,  and  back  again  to  a  common 
fraction. 

3.  Reduce  -{^  to  a  decimal,  and  back  again  to  a  common 
fraction. 

4.  Reduce  |^  to  a  decimal,  and  back  again  to  a  common 
fraction. 

5.  Reduce  f  to  a  decimal,  and  back  again  to  a  common 
fraction. 

6.  Reduce  Y^^xy  to  a  decimal,  and  back  again  to  a  common 
fraction. 

7.  Reduce  .53  to  a  common  fraction. 

8.  Reduce  .46  to  a  common  fraction. 

9.  Reduce  .325  to  a  common  fraction. 

10.  Reduce  .24  to  a  common  fraction. 

11.  Reduce  y^y  to  a  decimal,  and  back  again  to  a  common 
fraction. 

12.  Reduce  |-g^  to  a  decimal,  and  back  again  to  a  common 
fraction. 


DECIMAL   FRACTIONS.  129 

186.  Model  of  a  Recitation. 

' .    Reduce  ^^  to  a  decimal  fraction. 

1  It  will  generally  be  sufficiently  ac- 

72)1.00(.014 —  curate  to  extend  the  quotient  only  to 

72  four  or  five   places    of  decimals,   and 

write  in  the  last  place  the  figure  that 

280  will  make  the  quotient  the  nearer  cor- 

288  rect,  with  ( — )  after  it  if  the  figure  be 

— —  too  large,  and  (-f-)  if  it  be  too  small. 

But  if  it  be  required  to  multiply  such  a 
quDtient,  some  allowance  should  be  made  for  its  incorrect- 
ness. 

2.  How  much  would  125  cords  of  wood  come  to,  at  $5.16 
per  cord? 

/^  ,A  5  times  6  are  30,  but  if  another  6  of  the 

1  Q  r         repetend  were  written  and  multiplied  by  5,  it 

would  aflford  3  to  be  added  to  this  30,  •  making 

33;  therefore,  write  3  in  the  first  place.     2 

2583         times  6  are  12,  but  if  another  6  were  written 

10333         ^^^  multiplied  by  2,  it  would  aflford  1  more  to 

51666        ^^  added  to  this  12,  making  13 ;  write  the  3, 

and,  since  this  3  is  a  repetend,  continue  it  to 

cbnA/r  Qo         the  lowest  place ;    so,  also,  continue   the   6 

down  to  the  lowest  place.     The  sum  of  the 

first  column  is  12,  but  as  there  might  be  a  lower  column  like 

this,  which  would  aflford  1  more  for  this  column,  write  3  in 

the  first  place,  &c. 

3.  What  would  be  the  cost  of  1  pair  of  boots,  if  5  pairs 
cost  $16  ? 

5  is  contained  3  times  in  16,  and  1 

5)  16.  ($3,333  -f-        remains,  which  reduce  to  tenths,  making 

15    •  with  the  .6  belonging  to  the  repetend, 

■'  16  tenths,  which  will  give  .3  for  the 

16  quotient;  1  tenth  remains,   which  re- 

15  duced  to  hundredths,  and  added  to  the 

.06,  will  give  .03  for  the  quotient,  &c. 

187.  Exercises  in  the  Use  of  Infinite  Decimals. 
In  like  manner  J  solve  and  explain  the  following  problems. 

1 .    If  there  are  $4  in  1  sterling  pound,  what  is  the  value 
of  1000  sterling  pounds  ? 


130 


ARITHMETIC. 


2.  If  $.16  make  a  shilling,  what  is  the  value  of  6 
shillings  ? 

3.  If  $.083  make  a  sixpence,  what  is  the  value  of  12  six- 
pences ? 

4.  If  $.0416  make  a  threepence,  what  is  the  value  of  24 
threepences? 

5.  What  is  the  .value  of  100  yards  of  silk,  at  $.83  per 
yard? 

6.  If  5  spelling  hooks  cost  $.83,  what  is  the  price  of  1  of 
vhem  ? 

7.  What  is  the  value  of  1  sterling  pound,  if  12  pounds 
make  $53.  ? 

8.  If  6  yards  of  silk  cost  $5,  what  is  that  per  yard? 

9.  If  24  oranges  cost  $1,  how  much  is  that  apiece? 

10.  What  would  1  comh  cost,  if  60  combs  should  cost 
$5.00  ? 

188.  Model  of  a  Recitation. 

1.  At  $.12  a  pound  for  raisins,  how  many  pounds  may  be 
bought  for  $2.88? 

As  many  pounds  may  be  bought 

.12)2.88(24  pounds.       as  $.12  is  contained  times  in  $2.88 ; 

24  that  is,  12  cents  in  288  cents,  or,  12 

hundredths  in  288  hundredths,  which 

48  will  be  as  many  times  as  12  things  of 

48  any  kind  is  contained  in  288  things 

of  the  same  kind;  that  is,  24  times ; 

therefore,  24  pounds  is  the  answer 
required. 

2.  If  a  charitable  person  distribute  3  barrels  of  flour  to  the 
poor,  giving  them  .125  of  a  barrel  apiece,  to  how  many 
persons  could  he  give  a  portion  ? 

To  as  many  persons  as  .125  is 

.125)3.000(24  persons.  contained  times  in  3. 

250  Observe,  that,  in  the  first  ex- 

ample)  both  divisor  and  dividend 

500  being  of  the  same  denomination, 
500,  the  quotient  was  an  integral 
number,  and  necessarily  so,  from 

the  fact,  that  if  both  divisor  and 
dividend  be  of  the  same  denomination,  it  cannot  affect  the 
quotient i  whether  that  denomination  be  pounds,  barrels^  miles. 


DECIMAL   FRACTIONS.  131 

ddtfSy  dollars,  cents,  mills,  tenths,  hundredths,  or  thousandths. 
Fo  r  instance  ;  6  things  of  any  denomination  are  contained  in 
12  things  of  the  same  denomination,  2  whole  times. 

Therefore,  if  this  dividend  be  reduced  to  the  same  denomi- 
naion  as  the  divisor,  that  is,  to  thousandths,  the  quotient 
so  far  must  be  an  integral  number.  The  quotient  being  24, 
so  many  persons  could  receive  a  portion  of  the  flour. 

3.  How  many  bushels  of  apples  at  $.5  per  bushel  may  be 
boightfor  $.375? 

As  many  bushels  as  .5  is  contain- 

.«»).375{.75  bushels.      ed  times  in  .375.     5  tenths  is  not 

35  contained  in  3  tenths  ;  therefore,  there 

can  be  no  units  in  the  quotient ;  write 

25  the  point,  and  take  into  consideration 

25  one  more  figure  of  the  dividend,  and 

the  quotient  figure  thence   obtained 

will  be    tenths,  since  tenths  follow 
next  to  units ;  then  come  hundredths,  &c.,  in  their  own  order. 

4.  At  $6.25  per  barrel  for  flour,  how  many  barrels,  or 
wliat  part  of  a  barrel  may  be  bought  for  $.03125  ? 

As  many  barrels  as  6.25  is 

6. 25).03125(.005  barrels.         contained    times   in   .03125. 

3125  The  divisor  being  hundredths 

'  only  the   hundredths  of  the 

^  dividend  will  aflford  units  for 

the  quotient,  but  625  (hundredths)  not  being  contained  in  the 
3  (hundredths,)  there  will  be  no  units  in  the  quotient.  Write 
the  point,  and  take  into  consideration  one  more  figure  of  the 
dividend,  which  gives  0  tenths  for  the  quotient;  the  next 
figure  gives  0  hundredths  for  the  quotient,  and  the  next 
figure  gives  5  thousandths,  which,  as  there  is  no  remainder, 
is  the  answer  required. 

189.    Observation. 

In  the  division  of  decimals  by  integral,  or  decimal  num- 
bers, you  need  have  little  difficulty  in  ascertaining  the  right 
place  for  the  point,  if  you  observe  under staiidingly,  that  the 
figures  of  the  dividend,  as  low  as  the  lowest  figure  of  the 
divisor,  and  no  farther,  loill  give  integral  quotient  figures  ;, 
and  if  you  are  careful  to  write  the  point  in  the  quotient  as 
soon  as  you  have  come  to  its  place.  Should  not  the  dividend 
already  be  as  low  as  the  divisor,  make  it  so  by  annexing 
ciphers. 


132  arithmetic. 

190«   Proof  of  the  Pointing  in  the  Division  of  Deci- 
mals. 

To  prove  whether  you  have  pointed  the  quotient  correctly, 
consider  that  the  divisor  and  quotient  being  the  two  factors 
of  the  dividend,  (75)  must  together  have  the  same  number  . 
of  decimal  figures  as  the  dividend,  (1T6).  If  your  work 
stands  this  test,  probably  you  have  put  the  point  in  its  right 
place. 

191.    Exercises  in  the  Division  qf  Decimals. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  How  many  umbrellas,  at  $1.25  apiece,  may  be  bought 
for  $3.75  ? 

2.  How  many  pairs  of  half-hose,  at  $.35  a  pair,  may  be 
bought  for  $1.40  ? 

8.    How  manv  pounds  of  coifee,  at  $.12  a  pound,  may  be 
bought  for  $13.44  ? 

4.  How  many  pounds  of  cheese,  at  $.07  a  pound,  may  be 
bought  for  $5.25  ^ 

5.  At  $1.50  per  yard,  how  many  yards  of  cassimere  may 
be  bought  for  $24.  ? 

6.  At  $.80  per  yard,  how  many  yards  of  kersey  may  be 
bought  for  $20  ? 

7.  At  $.40  per  yard,  how  many  yards  of  flannel  may  be 
bought  for  $12? 

8.  At  $.20  per  yard,  hov/  many  yards  of  calico  may  be 
bought  for  $32  ? 

9.  If  $108.50  be  paid  for  cassimere,  at  $4  per  yard,  how 
many  yards  were  bought  ? 

10.  If  $18.50  be  paid  for  broad-cloth,  at  $5  a  yard,  how 
many  yards  were  bought? 

11.  If  $28.35  be  paid  for  4.5  barrels  of  flour,  how  much 
is  that  a  barrel  ? 

12.  If  $153,525  be  paid  for  26.7  cords  of  wood,  what 
would  1  cord  cost  ? 

13.  What  would  1  bushel  of  wheat  cost,  if  14.75  bushels 
cost  $18.4375  ? 

14.  What  would  lib.  of  sugar  cost,  if  375.6  pounds  cost 
$46.95? 

15.  What  would  1  ton  of  potash  cost,  if  28.75  tons  cost 
$3616.175? 


COMPOUND   NUMBERS.  133 

16.  What  would  1  bushel  of  corn  cost,  if  63.5  bushels 
cc  St  $49.53? 

17.  What   costs    1   yard    of    cloth,   if    79.4  yards   cost 
$187,384  ? 

18.  How  much  sugar,  at  $.125  per  pound,  can  be  bought 
for  $15.50? 

19.  If  112  pounds  of  iron  cost  $7.28,  what  is  the  cost  of  1 
pcund? 

20.  How  many  times  is  $.06  contained  in  $33.60  ? 

21.  How  many  times  is  .46  contained  in  18.4? 

22.  How  many  times  is  .18  contained  in  7.02? 

23.  How  many  times  is  .4  contained  in  2.5  ? 

24.  Divide  13.2  by  17.6. 

25.  Divide  61.512  by  2.4. 

26.  Divide  .063  by  10.5. 

27.  Divide  1.8144  by  10.5. 

28.  Divide  5.38575  by  1.075. 

29.  What  part  of  8  is  3? 

30.  What  part  of  8  is  .3? 

31.  What  part  of  2.4  is  .6? 

32.  What  part  of  10.35  is  5.175? 

33.  How  many  times  is  .2  contained  in  .06  ? 

34.  How  many  times  is  .04  contained  in  .008? 

35.  Divide  .00003  by  .003. 

36.  Divide  .000011021  by  .0107. 

37.  Divide  .00001  by  .025. 

38.  Divide  47  by  .1. 

39.  Divide  3.  by  .0003. 

40.  What  part  of  1.006  is  .002012? 


VIII.    COMPOUND    NUMBERS- 

19^.   Compound  Numbers  defined  and  illustrated. 

In  simple  numbers,  such  as  we  have  heretofore  employed, 
the  several  orders  of  units  increase,  or  decrease,  by  the 
uniform  ratio  10  or  yV,  that  is,  one  unit  of  each  order  equals 
10  units  of  the  next  lower  order  (IO5)  or  yV  of  a  unit  of  the 
next  higher  order.  But  a  compound  number  is  a  number  which 
expresses  a  quantity  in  several  denominatiozis  having  no  uni- 
form ratio.  Thus,  4  yards  2  feet  6  inches,  is  a  compound 
number ;  12  inches  making  a  foot,  and  3  feet  a  yard. 
12 


134  ARITHMETIC. 

The  ratios  (26^)  of  the  denominations  of  compound  num" 
hers,  are  exhibited  in  the  folhioing  tables^  which  should  be 
thoroughly  committed  to  memory. 

193.   Long  Measure  illustrated. 

Long  measure  is  used  in  measuring  distances  between  two 
points.  Its  unit  is  the  mile,  which  is  divided  and  subdivided 
according  to  the  following 

Ta])le. 

m.  fur.  rods.  yds.  ft.  in. 

1  =  8  =  320  =  1760  =  5280  ==  63360. 

1  =  40  =  220  =  660  =  7920. 

1=   51=   161=   198. 

1  =   3  =   36. 


1  =   12. 

40  rods     ==  1  furlong. 
8  fur.        =  1  mile. 


12  inches  =  1  foot. 
3  ft.        =1  yard. 
5\  yd.     =  1  rod. 

194.    Cloth  Measure  illustrated. 

Cloth  measure  is  used  in  measuring  chths^  laces,  ribbons, 
^c.  Its  unit  is  the  yard  of  long  measure,  which  is  divided, 
and  subdivided,  according  to  the  following 


Table. 

yd.        qr.          na.            in. 

1  =  4  =  16  =  36. 

21  inches  =  1  nail. 

1=    4=    9. 

4  na.       =1  quarter. 

1=    21. 

4  qr.        =1  yard. 

19S«    Square  Measure  illustrated. 

Square  measure  is  used  in  measuring  surfaces.  The  unit  of 
measure  is  a  square,  which  is  a  plain  surface  having  four  equal 
sides  and  angles  (439).  It  is  called  a  square  inch,  foot, 
yard,  &;c.,  according  as  its  side  is  one  inch,  foot,  yard,  &c.,  in 
length. 

If  lines  one  inch,  or  foot,  See,  apart,  be  drawn  parallel 
(4S9)  to  two  opposite  sides  of  a  square,  and,  in  like  manner, 
lines  be  drawn  parallel  to  the  other  two  sides,  the  number  of 
squares  thus  formed,  or  the  superficial  contents  of  the  square, 
will  be  the  second  power  (49)  of  the  inches,  or  feet,  fee,  in 
a  side  of  the  square ;  for  the  number  of  inches,  or  feet,  &c., 
in  a  side  of  the  square,  will  be  equal  to  the  number  of  rows 
of  squares,  and  to  the  number  of  squares  in  each  row,  the 


COMPOUND   NUMBERS.  136 

product  of  which  will  express  the  contents  of  the  square. 
Taus,  a  square  foot  would  make  12  rows  of  12  square 
inches  each,  equal  to  144  square  inches,  which  is  the  second 
pc'Wer  (363)  of  12,  the  number  of  inches  in  the  side  of  a 
square  foot. 

Hence,  the  contents  of  any  rectangular  surface  (439)  is 
the  product  of  its  length  and  breadth. 

The  square  m%le  is  divided  and  subdivided  according  to  the 
following 

Table, 

n.    acres,     roods.        rods.  yds.  ft.  in. 

1=640=2560=102400=3097600  =27878400  =4014489600. 

1=      4=       160=       4840  =      43560  =       6272640. 

1=        40=       1210  =       10890  =       1568160. 

1=  30i=  272i=  39204. 


144  inches  =  1  foot. 
9  ft.        =1  yard. 
304  yd.     =  1  rod. 


1  =  9  =  1296. 

1  =  144. 

40  rods     =  1  rood. 

4  roods  =  1  acre. 

640  acres  =  1  mile. 


196.   Cubic  Measure  illustrated. 

Cubic  measure  is  used  in  measuring  solids  and  capacities^ 
01  anything  that  has  three  dimensions,  length,  breadth,  and 
thickness. 

The  unit  of  measure  is  a  cube,  which  is  a  solid  having  six 
equal  square  faces.  (439).  It  is  called  a  cubic  inch,  or  foot, 
&c.,  according  as  a  side  of  a  face  of  it  is  one  inch,  or  foot, 
&c.,  in  length. 

If  12  boards,  each  a  foot  square  and  one  inch  thick,  be 
piled  together,  they  would  make  a  cubic  foot;  but  each 
board  maybe  divided  into  12  equal  pieces  one  inch  wide  and 
thick,  and  each  piece  into  12  cubic  inches ;  therefore,  each 
board  would  make  12  X  12  =  144  cubic  inches,  and  the  12 
boards,  or  cubic  fo  Dt,  would  make  12  X  12  X  12  ==  1728 
cubic  inches. 

Hence,  the  conte?its  of  a  cube  is  the  product  of  its  three 
dimensions,  or  the  product  of  its  base  (439)  and  height,  or 
the  third  power  (49)  of  a  side  of  onje  of  the  cube's  faces. 

Hence  also,  the  contents  of  any  solid,  SfC,  having  rectan^ 
gular  faces,  (439)  is  the  product  of  its  three  dimensions, 

A  cubic  yard  is  divided  and  subdivided,  according  to  the 
following  table.  ,.  '^ 


136  ARITHMETIC. 

Table. 

yd.         ft. 


1  =  27  =  46656. 
1=    1728. 


1728  inches  =.  1  foot. 
27  ft.         =  1  yard. 
128  feet  make  1  cord ;  but  it  is  usual  to  consider  |-  of  a 
cord,  or  16  cubic  feet,  1  cord-foot ;  hence,  8  cord-feet  make  1 
cord. 

50  feet  of  timber,  make  1  ton.  But  a  ton  of  round  timber 
will  make  only  40  feet  of  square  timber,  as  ^  is  allowed  for 
waste  in  squaring. 

107*   Dry  Measure  illustrated. 

Dry  measure  is  used  in  measuring  grain,  fruit,  salt,  and 
similar  dry  goods.  Its  unit  is  the  bushel,  which  is  divided, 
and  subdivided,  according  to  the  following 

Table. 

"     bu.        pk.        gal.  qt.  pt. 

1  =  4  =  8  =  32  =  64. 
1  =  2=    8  =  16. 

1=    4=    8. 
1=2. 
One  gallon  contains  268|-  cubic  inches  (190.) 

198«   Liquid  Measure  illustrated. 

Liquid  measure  is  used  in  measuring  all  kinds  of  liquids. 
Its  unit  is  the  gallon,  which  is  divided,  and  subdivided 
according  to  the  following 

Table. 

gial.        qt.         pt.  gill. 

1  =  4  =  8  =  32.      4  gills  =  1  pint. 
1  =  2  =:    8.      2  pt.     =1  quart. 
1  =4.      4  qt.     =:  1  gallon. 
One  gallon  contains  231  cubic  inches. 
One  gallon  of  milk,  and  malt  liquors  contains  282  cubic 
inches. 

199.   Troy  Weight  illustrated. 

Troy  weight  is  used  in  weighing  precious  metals,  and 
liquids.  Its  unit  is  the  pound,  which  is  divided,  and  sub- 
divided, according  to  the  following 


2  pints  =  1  quart. 
4  qts.    =  1  gallon. 
2  gals.  =  1  peck. 
4  pks.  =:  1  bushel. 


COMPOUND   NUMBERS. 


137 


TabU. 


oz.  dwt.  gr. 

12  =  240  =  5760. 
1=   20=   480. 
1  =     24. 


24  grains  =  1  pennyweight. 
20  dwt.     =  1  ounce. 
12  oz.        =  1  pound. 


SDO*   Apothecaries'  Weight  illustrated. 

Apothecaries^  weight  is  used  in  compounding  medicines, 
lis  unit  is  the  pounds  which  is  divided,  and  subdivided,  accord- 
irg  to  the  following 


TahUi 

1  =  12  =  96  =  288  =  5760. 

1=   8=   24=   480.' 

1=     3=     60. 

1=     20. 


20  grains  =  1  scruple. 

3  9         =1  dram. 

8  3         =1  ounce. 
12  §         =:  1  pound. 


901.  Avoirdupois  Weight  illustrated. 

Avoirdupois  weight  is  used  in  weighing  coarse  goods^  such 
as  are  not  weighed  by  the  Troy,  or  Apothecaries*  weight.  Its 
unit  is  the  ton^  which  is  divided,  and  subdivided,  according  to 
the  following 

Table, 


ton       lb.  oz.  dr.  gr. 

1  =  2000  =  32000  =  512000  ==  14000000. 
1=        16=       256=         7000. 
1==  16=  437i 

1=  27H. 


27H  grs.  =  1  dram. 
16  dr.      =  1  ounce 
16  oz.    ,=  1  pound. 
2000  lb.      =  1  ton. 


The  hundred-weight  and  quarter  are  generally  dispensed 
with ;  and  2240  pounds  are  no  longer  considered  a  ton. 

The  grain  in  the  three  weights  is  the  same,  but  the  other 
denominations,  though  agreeing  in  name,  differ  in  weight, 
excepting  in  Troy  and  Apothecaries'  weight,  where  they  are 
identical. 

The  weight  of  anything  together  with  the  container,  is 
called  gross  weight ;  and  the  remainder,  after  deduction  has 
been  made  for  the  container,  &c.,  is  called  v£t  weight, 
(307.) 

12=^    . 


4     # 


vj8  arithmetic. 

202.   Time  illustrated. 

Ti7?ie  is  divided  into  years  by  the  revolutions  of  the  earth 
about  the  sun,  and  years  into  days  by  the  revolutions  of  the 
earth  upon  its  axis.  A  year  is  divided,  and  subdivided,  ac- 
cording to  the  following 

Table, 
Year  days,  hours,  minutes,  seconds. 

1  =  3654  =  8766  =  525960  =  31557600 

1  ==   24  =   1440  =    86400 

1  =    60  =    3600 

1  =      60 


60  seconds  ==  1  minute, 
60  m.    =1  hour, 


24    h.  =  1  day, 
365 J  d.  =  1  year. 


365 J  days,  though  not  exactly  a  year,  is  sufficiently  ac- 
curate for  ordinary  purposes,  and  will  be  considered  a  year, 
unless  it  be  otherwise  specified.  It  is  usual  in  calendars  to 
reckon  365  days  to  all  years,  except  those  divisible  by  4,  to 
which  366  days  are  allowed ;  but  centennial  years,  though 
divisible  by  4,  have  only  365  days,  except  the  years  which 
are  divisible  by  400,  which  have  366  days.  Years  having 
366  days  are  called  leap  years,  in  which  February  has  29 
days,  otherwise,  only  28.  The  calendar  months,  April,  June, 
September,  and  November,  have  each  30  days  ;  and  January, 
March,  May,  July,  August,  October,  and  December,  have 
each  31  days.  But  in  calculations  involving  dates,  30  days 
are  considered  a  month,  and  12  months  a  year. 

303     Circular  Measure  illustrated. 

Circular  Measure  is  used  in  measuring  circles,  (-ASOj)  and 
their  circumferences,  particularly  angles,  latitude  and  longi- 
tude, and  the  relative  situations  of  the  heavenly  bodies. 

If  from  the  centre  of  a  circle^  straight  lines  be  drawn,  divid- 
ing the  circumference  into  360  equal  parts,  or  arcs,  each  of 
these  arcs  is  called  a  degree,  as  is  also  each  of  the  spaces 
comprehended  by  two  of  the  straight  lines,  or  radii. 

Hence,  a  degree  being  -^^  of  a  circle,  or  of  its  circum- 
ference, its  extent  will  be  greater,  or  less,  according  to  the 
size  of  the  circle. 


H 


COMPOUND   NUMBERS.  139 


The  divisions,  and  subdivisions,  of  a  circle  and  its  circwmr 
firence  are  exhibited  in  the  following 

I  Table. 

C.        signs,      degrees,       minutes,  seconds. 

1  =  12  =  360  =  21600  =  1296000 

1  ==    30  =     1800  =  108000 

1  ==        60  =  3600 


1  =  60 


<)0"  (seconds)  =  1  minute, 
00'  =1  degree, 


30°  =  1  sign, 
12s.  =  1  circle. 


S<l>4«    English  Money  illustrated. 

English  Money  is  the  national  currency  of  Great  Britain. 
It  was  the  currency  of  the  United  States  till  the  establishment 
of  Federal  Mon£y,  in  1786,  and  is  partially  used  here  at  pres- 
ent. Its  unit  is  the  pound,  which  is  difided,  and  subdivided, 
according  to  the  following 

Table. 


£. 

s. 

d. 

qr. 

1  = 

20 

= 

240  = 

960 

1 

= 

12  = 
1  = 

48 
4 

4  farthings  =  1  penny. 
12  pence       =  1  shilling.. 
20  shillings  =  1.  pound. 


SCItl.    Currencies  of  English  Money  illustrated. 

The  term  pound  represents  different  values  in  the  differ- 
ent currencies ;  so  also  do  the  other  denominations  of 
English  money,  according  to  the  following 

Table, 
£1,  Sterling,      =S4|,   used  in  England. 
£1 ,  Can.     Cur,  =  $4,        *'     "  Canada  and  Nova  Scotia. 
£1,  N.  E.      "   =S3i,      "     "  N.E.,Va.,Ky.,andTenn. 
£1,  N.  Y.      "   =  $2J,      "     "  N.  Y.,  Ohio,  and  N.  C. 
£1,  Penn.      "=$2|,      ''     "  Penn.,N.J.,Del.andMd. 
£1,  Georgia  "    =i4f,      '*     "  Ga.  and  S.C. 
$1  =  4^.  6^.       =  £^V,  Sterling. 
$1==55.  =£|j    Can.  Currency. 

$1  =  6^.  =£.3,  N.E.       " 

$1  =  8^.  =£.4,  N.Y.      " 

$1  =  7^.6^.       =£f,    Penn.      " 
$1  =  4^.8^.       =£^,Ga. 
M.84  is  the  present  legal  value  of  the  pound  sterling. 


140 


ARITHMETIC. 


SS06.   New  England  Currency  illustrated. 

Neio  England  Currency  is  still  much  used  in  appraising 
articles  of  merchandise.  The  most  common  prices  are  ex- 
hibited, reduced  to  Federal  money,  and  aliquot  parts  of  a 
dollar,  in  the  following 

Table. 


S.  d. 

S.  d. 

0  3    = 

0  4i= 

S.04^  = 
.06J  = 

3  3  = 
3  6  = 

$  .54i  = 

.581  = 

ta 
^ 

0  6    = 

.081  = 

tV 

3  9  = 

.62^  = 

1 

0  9    = 

.121  = 

i 

4     = 

.66|  = 

1 

1        = 

.16|  — 

i 

4  3  = 

.70i  = 

H 

1  3    = 

•20i  = 

■ix 

4  6  = 

.75    = 

J 

1  6    = 

.25    = 

i 

4  9  = 

.79i  = 

if 

19    = 
2       = 
2  3    = 

.29i  = 
.331  = 
•37i  = 

5      = 
5  3  = 
5  6  = 

.831  = 
•87*  = 
.91*  = 

1 

2  6    = 

2  9    = 

3  = 

.41f  = 
•45i  = 
.50    = 

A 

¥ 

5  9  = 

6  = 

.95f  = 
1.00    = 

¥ 

307.   Model  of  a  Recitation. 

1.    How  many  inches  long  is  a  road,  which  measures  3 
miles,  6  furlongs,  25  rods,  2  yards  and  1  foot  ? 

Since  there  are  8  fur- 
3m.  6f.  25rds.  2yds.  1ft. 
8 


30  furlongs. 
40 


1225  rods. 
11 


longs  in  a  mile,  there  will 
be  8  times  as  many  fur- 
longs as  miles  ;  8  times 
3  are  24,  which,  together 
with  the  6,  make  30  fur- 
longs. Since  there  are 
40  rods  in  a  furlong, 
there  will  be  40  times  as 
many  rods  as  furlongs ; 
40  times  30  are  1200, 
which,  together  with  the 
25,  make  1225  rods. 
Since  there  are  5|,  or 
-y-,  yards  in  a  rod,  there 
will  be  -y-  as  many  yards 
as  rods  ;  ^  of  1225  are 
6737 J,  which,  together  with  the  2,  make  6739 J  yards.    Since 


2)13475  half-yards. 
6739^  yards. 


20219i  feet. 
12' 


242634  inches. 


COMPOUND   NUMBERS.  141 

th'jre  are  3  feet  in  a  yard,  there  will  be  3  times  as  many  feet 
as  yards  ;  3  times  6739^  are  20218J,  which,  together  with 
th3  1,  make  20219|  feet.  And,  since  there  are  12  inches  in 
a  bot,  there  will  be  12  times  as  many  inches  as  feet ;  12  times 
2C2l9i  are  242634  inches,  which  is  the  answer  required. 
2.    Kediice  242634  inches  to  higher  denominations. 

Since  it  takes 

12)  242634  inches.  J^/''?^'  ^°'n 

' loot,  there   will 

3)  20219  ft.  6  inches.  "f  f^  ^?  ^^^^ 

' leet  as  mcnes  ; 

ri        iiN«^QQ    A     oA  1^     of    242634 

^  =  V)  6739yds.  2ft.  Se  20219  feet, 

and  6  inches  re- 

ll)1347ahalf.yards.  T^  It:. 

40)  1225  rods,  3  half-yards.  ^^  ^f  ,^there 

,    8)  30  fur.  25  rods.  ^7^720211 

3,n.6fur.25rds.2yds.lft.       T/PLTi 

maining.  Since 
it  takes  5|,  or  -^^,  yards  for  1  rod,  there  will  be  as  many  rods 
as  -y-  is  contained  times  in  6739  ;  multiply  by  2,  to  ascertain 
how  many  times  \  is  contained,  (IS^,)  and  divide  that  pro- 
duct by  11,  to  ascertain  how  many  times  -y-  is  contained, 
which  gives  1225  rods,  and  3  half-yards  remaining.  Since 
it  takes  40  rods  for  1  furlong,  there  will  be  ^  as  many  fur- 
longs as  rods  ;  ^j^  of  1225  are  30  furlongs,  and  25  rods  re- 
maining. And,  since  it  takes  8  furlongs  for  1  mile,  there 
will  be  ^  as  many  miles  as  furlongs  ;  -|-  of  30  are  3  miles, 
and  6  furlongs  remaining :  making,  in  all,  3  miles,  6  furlongs, 
25  rods,  2  yards,  and  1  foot,  which  is  the  answer  required. 
The  2  yards  are  obtained  thus  :  1  of  the  2  feet,  and  the  6 
inches,  make  \  yard,  which,  with  the  f  yards,  make  f  =  2 
yards. 

308.    Reduction  defined. 

Reduction  is  the  changing  of  a  compound  number  mto  a 
simple  number  of  the  same  value,  as  in  the  first  example, 
(•JOTj)  or  the  changing  of  a  simple  number  into  a  compound 
number  of  the  same  value,  as  in  the  second  example ;  or,  the 


1^  ARITHMETIC. 

changing  of  a  nurriber  of  any  kind  into  another  of  the  same 
value,  as  there  are  frequent  examples  in  this  book. 

S09.   Observation. 

Observe,  (Q07^)  thatj  in  the  first  example,  the  simple 
number  of  the  highest  denomination  ^7^  the  given  compound 
number,  is  reduced  to  the  next  lower  denomination,  to  which 
is  added  what  there  may  be  of  this  loioer  denomination  ;  that 
this  sum  is  reduced  to  the  next  lower  denomination  still,  and 
increased  as  before,  and  so  on,  till  all  is  reduced  as  loio  as 
desired :  the  reduction  in  each  case  being  performed  by  mul- 
tiplying by  the  number  which  expresses  how  many  units  of  the 
next  lower  denominMion  make  a  unit  of  the  simple  number  to 
be  reduced* 

Observe,  also,  that,  in  the  second  example,  the  number  in 
each  denomination  is  divided  by  the  number  which  expresses 
how  many  units  of  its  own  denomination  make  a  unit  of  the 
next  higher  derwmination  ;  that  the  last  quotient,  together  with 
the  several  remainders,  form  the  compound  number  required, 

310*   Exercises  in  the   Keduction  of  Compound  Num- 
bers. 

Solve  and  explain  the  following  problems,  on  the  left,  like 
the  first,  and  those  on  the  right,  like  the  second,  of  the  above 
examples  (207). 

Long  Measure. 


1.  Reduce  6  m.  3  fur.  20 
rds.  3  yds.  and  2  ft.  to  feet. 

3.  How  many  rods  in  25 
miles  ? 


2.  Reduce  34001  feet  to 
higher  denominations. 

4.  How  many  miles  in 
16000  rods  ? 


1 .  Reduce  27  yds.  1  qr.  and 
3  n.  to  nails. 

3.  What  will  54  yds.  3  qrs. 
of  cloth  cost,  at  $.25  per  yard  ? 


311.   Cloth  Measure. 

2.  Reduce  3627  inches  to 
higher  denominations. 

4.  How  much  cloth  may 
be  bought  for  $75.25,  at  $.25 
per  nail  ? 


313.   Square  Measure. 


1.  How  many  rods  in  a 
pasture  which  measures  2  m. 
320  acres,  and  80  rods  ? 


2.   Reduce  1548800  yards 
to  acres. 


COMPOUND   NUMBERS. 


143 


3.  What  will  1  acre  and 
20  rods  of  land  cost,  at  $.05 
pe  r  foot  ? 


4.  How  much  land  may 
be  bought  for  $1575.25,  at 
$.05  per  foot  ? 


213«   Cubic  Measure. 


1.  Reduce  4  yds.  15  ft.  and 
144  inches  to  inches. 

3.  How  many  inches  in  3 
toils  of  timber  ? 

5.  What  cost  5  cords  and 
3  cord-feet  of  wood,  at  $.75 
per  cord-foot  ? 


2.  Eeduce  81648  inches  to 
higher  denominations. 

4.  How  much  timber  in 
267840  inches  ? 

6.  How  much  wood  may 
be  bought  for  $79.20,  at  $.75 
a  cord-foot  ? 


314.   Dry  Measure. 


1.  Reduce  5bu.  3pks.  and 
1  iial.  to  quarts. 

3,  What  cost  2bu.  Ipk. 
ar  d  3  qts.  of  chestnuts,  at  $.04 
per  quart  ? 


2.  How  many  bushels  in 
10752  cubic  inches  ? 

4.  What  would  be  the 
price  per  bushel,  if  $60  be 
paid  for  1920  pints  ef  shag- 
barks  ? 


SltS*    Liquid  Measure. 


1.  How  many  pints  in  25 
gals.  3  qts.  ? 

3.  What  would  a  milk- 
man receive  for  300  cans  of 
milk,  each  holding  2  gals,  and 
2  qts.,  at  $.05  per  quart  ? 


to 


2.   Reduce  1732   gills 
higher  denominations. 

4.  How  many  gallons  of 
molasses  in  a  cask  gauging 
7238  cubic  inches  ? 


216.   Troy  Weight. 


1.  In  7  lb.  11  oz.  3dwt.  9 
grs.  how  many  grains  ? 

3.  What  will  lib.  10 oz. 
15  dwt.  20  grs.  of  jewelry  cost, 
at  $.04  per  grain  ? 


2.  Reduce  45681  grains  to- 
higher  denominations. 

4.  How  much  jewelry  may 
be  bought  for  $975.20,  at 
$.02  per  grain  ? 


217.   Apothecaries'  Weight. 

1.   In  lib.  7§,  25,  19,  12  I      2.   Reduce  9876  grains  to 
gis.  how  many  grains  ?  |  higher  denominations. 


144 


ARITHMETIC. 


3.   Reduce  61b.  lOg,  75, 
29,  IGgrs.,  to  grains. 


4.  In  39S36  grains,  how 
many  pounds  ? 


218.    Avoirdupois  Weight. 


1.  Reduce  2  tons,  1200  lbs. 
13  oz.,  to  ounces. 

3.  What  cost  20  tons, 
500  lbs.  of  hay,  at  $.0075  per 
pound  ? 


2.  How  many  tons,  &c.,  in 
1539000  drams  ? 

4.  How  much  hay  may  be 
bought  for  $34.05,  at  $.005 
per  pound  ? 


219.   Time. 


1.  How  many  seconds  old 
IS  a  boy  who  has  lived  12  y. 
90  d.  15  h.  20  m.  30  s.  ? 

3.  If  a  clock  tick  60  times 
a  minute,  how  many  times 
would  it  tick  in  16  years  ? 


2.  How  many  days  has  a 
child  lived,  whose  age  is 
31536000  seconds? 

4.  How  long  has  a  watch 
run,  whose  minute  hand  has 
turned  round  46975  times  ? 


290.    Circular  Measure. 


1.    Reduce  1  sign,  15°  to 
seconds. 

3.  If  Massachusetts  ex- 
tends 3°  4r  in  longitude, 
what  is  its  extent  in  seconds  ? 


2.  Reduce  749408  seconds 
to  higher  denominations. 

4.  What  is  the  extent  of 
Massachusetts  in  latitude,  it 
being  5940  seconds  from 
south   to   north  ? 


221.  English  Money. 


1.  Reduce  £25  10s.  6d. 
3qrs.  to  farthings. 

3.  In  £125  15s.  Canada 
currency,  how  much  Federal 


to 


2.   Reduce      9750qrs 
higher  denominations. 

4.  In  9600qrs.  N.  Y.  cur- 
rency, how  much  Federal 
money  ? 


money  f 

222.   Model  of  a  Recitation. 

1.    Reduce  £^^  to  farthings. 

There  will  be  20 
tA  vtSt4  =  ^%^  ^^s*  =  ^^^§  9[^s*  times  as  many  shil- 

lings as  pounds,  or 
f-H^'  >  12  times  as  many  pence  as  shillings,  or  ^^^. ;  and 


COMPOUND  MUMBERS.     >  145 

4  imes  as  many  farthings  as  pence,  or  ^^^  qrs.  =  166|  qrs., 
wliich  is  the  answer  required. 

2.    Reduce  £y®2ir  to  shillings,  pence,  and  farthings. 

Perform  the  multiplications  by  dividing 

S6)83(2s.  the  denominator,  (IO65)  or  divisor,  when 

72  practicable.     Thus : 

3)TI(3d.  There  will  be  20  times  as  many  shillings 

9  as   pounds,    or   T¥(T=2-Ty  =  ft  =  ^^s.     -J-^ 

-Q  shillings  will  make  12  times  as  many  pence, 

4  ii=T^  —  -V-  =  ^i^-     §  P^^^y  ^i^l  ^^^^  ^ 

qTq/02  nro  ^'^^^^  ^^  ^^^y  farthings,  f^  =  .|  =  2|  qrs. 

^)»i^t  qrs.  j^  ^^^^  2s.  3d.  2|qrs.,  the  answer  required. 

6 

"2 
2 

"0 

3^I3.     EXEECISES    IN    REDUCING    A    FRACTION    TO    UNITS    OP 
LOWER    DENOMINATIONS. 

In  like  Trmmver^  solve  and  explain  the  following  problems, 

1.  Reduce  £t^tj  ^^  *^^  fraction  of  a  farthing. 

2.  How  many  shillings  and  pence  in  |  of  a  pound  ? 

3.  Reduce  -^^^-^  of  a  mile  in  length  to  the  fraction  of  a 
rod. 

4.  What  is  the  value  of  f-  of  a  mile  ? 

5.  What  fraction  of  a  rod  is  yyV"?  ^^  ^^  ^^^®  • 

6.  What  is  the  value  of  |^  of  an  acre  ? 

7.  Reduce  ^^^  lb.  Troy  to  the  fraction  of  an  ounce. 

8.  Reduce  |^  of  a  Troy  pound  to  ounces,  pennyweights, 
and  grains. 

9.  What  fraction  of  a  grain  is  ^^^  of  an  ounce.  Apotheca- 
ries' weight  ? 

10.  Reduce  /^S  ^^  drams,  scruples,  and  grains. 

11.  Reduce  ^f^  of  a  pound.  Avoirdupois  weight,  to  the 
fraction  of  a  dram. 

12.  Reduce  fy  of  a  ton  to  lower  denominations. 

13.  What  fraction  of  a  quart  is  yf^  of  a  bushel  ? 

14.  Reduce  |-  of  a  bushel  to  quarts. 

15.  Reduce  2-^  of  a  gallon  to  gills. 

16.  How  many  quarts,  pints,  &c.,  in  ^  of  a  gallon? 

17.  How  many  cubic  inches  in  ^sizB  of  a  yard  ? 

18.  Reduce  -^  of  a  cubic  yard  to  inches. 

13 


146 


ARITHBIETIC. 


19  What  fraction  of  a  day  is  y/ft  ^f  a  year  ? 

20.  What  is  the  value  of  ^jj  of  a  day  ? 

21.  Reduce  -^^  of  a  degree  to  seconds. 

22  Reduce  J  of  a  circle  to  lower  denominations. 

3^4*   Model  of  a  Recitation. 

1.  What  part  of  a  pound  is  ^^Q-  qrs.  ? 

There  will  be  j^  as  many  pence  as 

^^aitgfe^iV?-         farthings,   or  \o^od.   (114,)    y'^    as 

many  shillings  as  pence,   or  ^fjs., 

and  2V  ^s  many  pounds  as  shillings,  or  Xyjj-,  the  answer 

required. 

2.  What  part  of  a  pound  is  4s.  6d.  ? 

There  will  be  12  times  as  many  pence  as  shillings,  and  6 
added,  giving  54d.,  and  there  will  be  ^^q  as  many  pounds  as 
pence,  which  (86)  gives  £^^^jj  =^  £-^j  as  required. 

3SS.   Exercises   in   reducing   lower  Denominations  to 
THE  Fraction  of  a  higher. 

I71  like  manner,  solve  and  explain  the  following  problems, 

1.  Reduce  f  d.  to  the  fraction  of  a  pound. 

2.  What  part  of  a  pound  is  10s.  6d.  ? 

3.  Reduce  |-  of  a  rod  in  length  to  the  fraction  of  a  mile, 

4.  Reduce  6  fur.,  26  rods,  11  feet,  to  the  fraction  of  a 
mile. 

5.  What  fraction  of  an  acre  is  t^tj  of  a  rod  ? 

6.  Reduce  3  roods,  13  rods,  90  feet,  to  the  fraction  of  an 
acre. 

7.  What  part  of  a  Troy  pound  is  |-  of  an  ounce  ? 

8.  Reduce  7  ounces,  4  dwt.,  to  the  fraction  of  a  Troy 
pound. 

9.  What  fraction  of  a  pound  is  ^  of  a  grain.  Apothecaries 
weight  ? 

10.'  Reduce  |^9  to  the  fraction  of  an  ounce. 

11.  What  fraction  of  an  ounce  is  35»  29,  10  grs.  ? 

12.  Reduce  |  of  a  pound  to  the  fraction  of  a  ton. 

13.  Reduce  1000  lb.  12  oz.  12  dr.  to  the  fraction  of  a  ton. 

14.  What  fraction  of  a  bushel  is  |  of  a  pint  ? 

15.  Reduce  3  pks.  1  gallon  to  bushels. 

16.  Reduce  J||  of  a  gill  to  gallons. 

17.  What  part  of  a  gallon  is  1  qt.,  1  pt.,  1  gill  ? 


COMPOUND   NUMBERS.  147 

18.  How  many  yards  in  -^  o(  a,  cubic  foot  ? 

19.  Reduce  81  cubic  inches  to  yards. 
20:,   What  part  of  a  year  is  24  hours  ? 

21.  Reduce  25  minutes,  30  seconds  to  days. 

22.  Reduce  §  of  a  minute  to  degrees. 

23.  What   part   of  a   circle   is  4  signs,  15  degrees,  15 
ninutes,  and  15  seconds? 

SSO.   Model  of  a  Recitation. 

1.    Reduce  .6  (183)  of  a  bushel  to  lower  denominations. 

.6  bushels.  There  will  be  4  times  as  many  pecks  as 

4  bushels,  or  2.6  pecks ;  the  fraction  .6  pecks, 

2.6  pecks.  will  make  2  times  as  many,  or  1.3  gallons  ; 

2  the  fraction  .3  gal.  will  make  4  times   as 

1.3  gallons.  many,  or  1.3  quarts ;  —  in  all,  2  pecks,  1  gal. 

4  1.3  qts.,  which  is  the  answer  required. 
1.3  quarts. 

SJ27.   Exercises  in  reducing  Decimal  Fkactions  to  units 
OF  A  lower  Denomination. 

In  like  manner ,  solve  and  explain  the  following  problems, 

1.  What  is  the  value  of  .625  of  a  bushel  ? 

2.  Reduce  .3125  of  a  mile  to  lower  denominations. 

3.  Reduce   .83  yards  to  lower  denominations. 

4.  Reduce  .375  acres  to  lower  denominations. 

5.  How  many  cubic  feet  in  .1875  of  a  cord? 

6.  How  many  quarts,  pints,  &c.,  in  .6  of  a  gallon  ? 

7.  What  is  the  value  of  .875§.  ? 

8.  What  is  the  value  of  .4  of  a  ton  ? 

9.  Reduce  .75  of  a  year  to  units  of  lower  denominations. 

10.  If  the  moon  advance  in  its  orbit  .406779661  +  of  a 
sign  in  a  day,  what  is  its  daily  advance  ? 

11.  Reduce  .625£  to  shillings  and  pence. 

2S8.   Model  Sf  a  Recitation. 

1.    Reduce   2  pks.    1    gal.  1.3  qts.  *to  the  decimal  of  a 
bushel. 


148  AEITHMETIC. 


The  1.5  quarts  will  make  ^  as  many 
1.3  quarts.  gallons,  that  is,  .3  gallons,  (ISS,)  which 
1.3  gallons.  annexed  to  the  1  gallon,  make  1.3  gallons; 
2.6  pecks.  1.3  gallons  will  make  |  as  many,  or  .6 

X"j      r~j  pecks,    which   annexed   to    the    2    pecks 

make  2.6  pecks ;  2.6  pecks  will  make  ^  as 
many,  or  .6  bushels,  which  is  the  answer  required. 

Or^  2pks.  1  gal,  1.3  qts.,  equal  to  21.3  qts.,  which  will 
make  -r}^  as  many  bushels,  or  s^j-^  of  a  bushel,  (9245  2,)  this 
common  fraction  reduced  to  a  decimal  fraction,  (ITS^)  makes 
.6  of  a  bushel  as  before. 

230*   Exercises  in  reducing  lower  Denominations  to  the 
Decimal  of  a  higher. 
In  like  manner^  solve  and  explain  the  following  prohleTns, 

1.  Reduce  1  pk.  1  gal.  1  qt.  1  pt.  to  the  decimal  of  a  bushel. 

2.  Reduce  3  fur.  15  rods,  2  yds.  to  the  decimal  of  a  mile. 

3.  Reduce  1  qr.  3  n.  2  in.  to  the  decimal  of  a  yard. 

4.  Reduce  25  yds.  3  ft.  72  inches  to  the  decimal  of  a 
square  rod. 

5.  Reduce  10  ft.  144  inches  to  the  decimal  of  a  ton  of  timber. 

6.  What  part  of  a  gallon  is  1  qt.  1  pt.  3  gills  ? 

7.  What  part  of  a  pound,  Troy,  is  1  oz.  1  dwt.  ? 

8.  What  part  of  a  ton  is  15001bs  ? 

9.  Reduce  January  to  the  decimal  of  a  year. 

10.  WTiat  part  of  a  revolution  will  a  person's  shadow  make 
in  1  hour,  if  its  hourly  motion  be  15  degrees? 

11.  Reduce  10s.  6d.  3qr.  to  the  decimal  of  a  pound. 

330.  Illustration  of  the  Mode  of  reducing  English 
Money  by  Inspection. 
Shillings,  pence,  and  farthings,  may  be  reduced  to  the 
decimal  of  a  pound  ;  or,  a  decimal  of  a  pound,  to  shillings, 
pence,  and  farthings,  more  expeditiously  by  inspection,  as 
follows. 

1.   Reduce  15s.  7d.  3qrs.,  to  the  decimal  of  a  pound. 

Since   20s.  =  1£;    2s.  = 

14s.  =.7£  /^£  =  ^i^  =  .l£;    therefore, 

Is.  =.05£  write    .1£  for  every  2  shil- 

7d.  3qrs.  =  .032  -f-  £      lings,  or  2  shillings  for  every 

.1£.     Thus,  14  of  the  15s. 

155.  7d.  3qrs.  =  .782  +  £       will  make  .7£. 


vr 


COMPOUND    NUMBERS.  149 


Since  ls.  =  ^£  =  -j-^^  £=:.05£,  write  .05£  for  an  odd 
shilling,  or  1  shilling  for  .05£.     Thus,  15s.  make  .75£. 

Since  24qrs.  =  ^^(s£  =  .025£,  add  1  to  the  farthings  for 

e\ery  24  farthings  in  the  given  pence  and  farthings,  and  they 

w  11  become  thousandths  of  a  pound ;  or  subtract  1  from  the 

thousandths  of  a  pound  for'every  25  thousandths  in  the  given 

I    thousandths,  and  the  remainder  wilLbe  farthings.     Thus,7d. 

I    3(^rs.  =  31qrs,,  which  will  be  a  trifle  more  than  .032£ ;  —  in 

f     all,  £.782 -f--  whi'^.h  is  the  answer  required. 

2J1.  Model  of  a  RecitatioxN. 

].    Reduce  £.782  to  units  of  lower  denominations. 

Since  .1£  is  2  shillings,  {^3^. 
.7£      =  14s.  ,7£  will  bR  .14  shillino-s.  ai>dtOi5£ 

.05£    =    Is.  being  1  shilling,  .75£  will  be  15 

.032£j= 7d.  3qrs.      shillings;  also,    .025£   being   S4 

7.g9-g — i.5g   7(i  3qrs,      farthings,    or    1    subtracted    from 

every  25  thousandths  leaving  far- 

things^   the  .032£   will  be  a  trifle  less    than   31qrs.  =:7d 

3(irs. ;  — in  all,  15s.  7d.  3qrs.,  which  is  the  answer  required. 

93S.    Observation. 

Observe,  (230,)  that  it  will  be^sujficiently  accurate,  in 
reducing  farthings  to  thousandths  of  a  pound,  to  add  1  to  the 
farthings  if  they  are  more  than  one  half  of  24,  or  2  if  they 
are  more  than  \\  times  24 ;  and,  in  reducing  thousandths  of 
a  pound  to  farthings,  (231 ,)  to  subtract  1  from  the  thou- 
sandths if  they  are  more  than  one  half  of  25 ;  or  2,  if  they 
are  more  than  \\  times  25;  siiice  in  either  case  the  error 
will  be  less  than  one  half  of  a  farthing. 

233.    Exercises    in    reducing    English  Money   by    In- 
spection. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  What  is  the  value  of  £.25  ? 

2.  Express  6s.  6d.  in  a  decimal. 

y^-^  3.  Reduce  £.625  to  shillings  and  pence. 

^        4.  Reduce  7s.  6d.  to  pounds. 

5.  Reduce  £.875  to  lower  denominations. 

6  Reduce  4s.  5d.  3qrs.  to  the  decimal  of  a  pound. 

7.  Reduce  £.323  to  units  of  lower  denominations. 

13*       j* 
/ 

i 

I' 


150  ARITH31ETIC. 

8.  Express  5s.  4d.  in  a  decimal  of  a  pound. 

9.  What  is  the  value  of  £.116  ? 

10.  Express  4|d.  in  pounds. 

11.  What  is  the  value  of  £.075. 

12.  Express  19s.  6d.  1  qr.  in  pounds. 

13.  What  is  the  value  of  £.48  ? 

14.  Express  12s.  lid., 3  qrs.  in  pounds. 

15.  Reduce  £.1875  to  shillings,  &c. 

16.  Reduce  Is.  2d.  3  qrs.  to  pounds. 

17.  Reduce  £.6  to  shillings,  &c. 

18.  Rpr!^7rr,  os.  Su=  CO  poimds. 

19.  Reduce  £12.18  to  pounds,  shillings,  pence,  and  fur 

'educe  £125  lis.  3d.  to  the  decimal  of  a  pound. 
934»   Model  of  a  Recitation. 

1.  If  from  a  piece  of  broad-cloth,  measuring  35  yds.  3  nls., 
a  tailor  cut  6  yds.  1  qr.  for  a  cloak  ;  3  yds.  3  qrs.  1  n.  for  a 
surtout ;  2  yds.  2  qrs.  2  nls.  for  a  frock-coat ;  1  yd.  2  qrs.  for 
a  pair  of  pantaloons ;  and  2  qrs.  2  nls.  for  a  vest ;  how  much 
cloth  would  he  use  for  these  garments  ? 

To  ^ascertain  how  much  cloth  he  would 
use,  you  must  add  together  the  several 
quantities  cut  off  from  the  piece. 

Arrange  together  the  numbers  to  be  added 
so  that  the  simple  numbers  of  each  denomi- 
nation may  stand  in  a  column  by  themselves. 
Add  the  numbers  of  each  denomination 
separately,  beginning  with  the  lowest.  The 
14     3     1  5  nails  of  the   first  column  are  equal  to  1 

nail,  which  write  in  its  own  column,  and  1 
quarter,  (lO^j)  vvhich  add  with  the  other  quarters,  making 
11  quarters,  equal  to  3  quarters, *which  write  in  their  own 
column,  and  2  yards,  which  add  with  the  other  yards, 
making  14  yards,  which  write  in  their  own  column, — making 
in  all,  14  yds.  3  qrs.  1  nl.,  which  is  the  answer  required. 

S33.    Observation. 

Observe,  (SSAj)  that^  in  the  addition  of  compound  iiuvi- 
hers,  the  amount  of  each  denomination  must  be  reduced  to 
units  of  the  next  higher  denomination,  a?id  added  there,  a?id 


yds. 

qrs. 

nia 

6 

1 

3 

3 

1 

2 

2 

2 

1 

2 

2 

2 

I 


COMPOUND   NUMBERS.  151 


tlat  in  each  column^  only  the  excess  over  exact  units  of  the 
mxt  higher  denomination  are  to  be  written. 

236.   Exercises  in  adding  Compound  Numbers. 

In  like  manner^  solve  and  explain  the  following  problems, 

1.  How  much  cloth  in  4  pieces,  measuring  as  follows  : 
25  yds.  3  qrs. ;  27  yds.  1  qr.  2  nls;  30  yds.  3  nls ;  and  28 
yards? 

2.  How  far  would  a  horse  trot  in  5  hours,  if  he  should 
bot  the  first  hour  12  miles,  3  fur.  25  rods ;  the  second  hour, 
1 1  miles,  7  fur.  20  rods ;  the  third  hour,  12  miles,  5  fur.  36 
r)ds,  5  yds.;  the  fourth  hour,  11  miles,  6  fur. ;  and  the  fifth 
liour,  13  miles,  15  rods? 

3.  If  by  one  road,  from  Lowell  to  Boston,  the  distance  be 
25  m.  2  fur.  and  20  rods,  and  by  another  road,  the  distance  be 
24  m.  7  fur.  12  rods  ;  how  much  distance  is  traveled  in  rid- 
ing to  Boston  by  one  road  and  returning  by  the  other  ? 

4.  How  much  land  in  a  farm  which  consists  of  50  acres, 
2  roods,  33  rods,  wood-land ;  25  acres,  14  rods,  mowing-land ; 
80  acres,  tillage ;  20  acres,  pasturing ;  12  acres,  1  rood, 
covered  with  water ;  and  10  acres,  25  rods,  swamp  ? 

5.  How  much  timber  in  two  sticks,  one  of  which  measures 
2  tons,  20  feet,  1642  inches ;  the  other,  1  ton,  15  feet,  1295 
mches  ? 

6.  How  much  wheat  does  that  man  raise^  who  has  three, 
fields,  and  raises  on  the  first,  45  bush.  3  pks. ;  on  the  second, 
36  bush.  1  pk.  7  qts. ;   and  on  the   third,    30  bush.    2  pks. 
1  quart? 

7.  How  much  molasses  in  two  casks  containing  as  follows  : 
70  gals.  3  qts.,  and  126  gals.  1  quart? 

8.  If  a  Johannes  weigh  18  dwts.  a  doubloon  16  dwts.  21 
grs.,  a  moidore  6  dwts.  18  grs.  and  an  English  guinea  5  dwt. 
6  grs. ;  what  is  the  weight  of  them  all  ? 

9.  If  an  apothecary  mix  of  one  kind,  7§,  55?  29 ;  of 
another  kind,  2§,  35  ;  and,  of  a  third  kind,  29,  10  g/s. ; 
what  is  the  weight  of  the  mixture  ? 

10.  If  a  load  of  hay  weigh,  without  the  wagon,  1  ton, 
1200  lbs.,  and  the  weight  of  the  wagon  is  1984  lbs. ;  what  is 
the  weight  of  the  whole  ? 

11.  If  the  load  of  hay  mentioned  in  tiie  last  problem,  were 
drawn  over  a  bridge  by  two  oxen  and  a  horse,  the  oxen 
weighing  1  ton,  187  lbs.,  the  horse  weighing  1160  lbs.,  and  the 


152  ARITHMETIC. 

driver  165  lbs.  12  oz. ;  how  much  more  did  the  bridge  sustain 
from  tills  team  passing  over  it  ? 

12.  If  £15  14s.  6d.  be  paid  for  a  pair  of  oxen,  £14  for  a 
horse,  and  £6  9s.  3d.  for  a  cow ;  what  would  be  the  whole 
cost? 

13.  How  old  would  a  man  be  when  his  eldest  child  is  12 
years,  25  days,  and  16  hours  old,  if  he  was  25  years,  344 
days,  and  10  hours  old,  at  the  birth  of  this  son  ? 

337.   Model  of  a  Eecitation. 

How  much  cloth  would  remain,  if  14  yds.  3  qrs.  1  nl.  be 
cut  from  a  piece  measuring  35  yds.  3  nails  ? 

To    ascertain   how    much    cloth    would 
yds.  qrs.  nis.  remain,  you  must  subtract  the  sum  of  what 

85     0     3  he  used  from  the  whole  piece. 

14     3     1  Write  the  subtrahend  under  the  minuend, 

placing  the  simple  numbers  of  each  denomi- 

20     1     2  nation  under  those  of  the  same  denomination. 

Beginning  with  the  lowest  denomination, 
take  1  nail  from  3  nails,  and  2  nails  remain,  which  write  in 
their  own  column  ;  reduce  1  of  the  35  yards  to  quarters, 
making  4  quarters,  (IO45)  from  which  take  the  3  quarters, 
and  1  quarter  remains,  which  write  in  its  own  column,  and 
take  14  yards  from  34  yards,  and  20  yards  remain,  which 
write  in  their  own  column,  —  making  in  all,  20  yds.  1  qr. 
2  nls.,  which  is  the  answer  required. 

S38.    Observation.  * 

Observe,  that,  in  subtraction  of  compound  numhersy  luhen 
a  number  of  any  denomination  in  the  minuend  is  less  than 
the  corresponding  number  in  the  subtrahend,  a  unit,  or  in 
some  cases,  a  part  of  a  unit,  of  a  higher  denomination  in  the 
minuend  must  be  reduced  to  make  up  the  deficiency, 

S30.    Exercises  in  subtracting  Compound  Numbers. 

In  like  manner,  solve  and  explain  the  folloioing  problems, 

1.  If  5  yds.  1  qr.  3  nls.  be  cut  from  a  piece  of  cloth 
measuring  20  yds.  3  qrs.,  how  much  would  remain? 

2.  What  is  the  difference  between  two  piles  of  wood,  one 
of  which  measures  15  cords,  3  cord-feet,  12  feet,  and  the  other, 
10  cords,  7  cord-feet,  6  feet  ? 


COMPOUND    NUMBERS.  153 

3.  If  a  farmer  raise  on  one  field  150  bush,  of  potatoes,  and 
or  another,  90  bush.  3  pks. ;  how  much  more  does  he  raise 
or  the  large  field  than  on  the  other. 

4.  If  a  merchant  draw  from  a  cask  of  molasses,  containing 
1£6  gals.  I  qt. ;  at  one  time,  13  gals.  3  qts. ;  at  another, 
1(  gals. ;  and  at  a  third  time,  25  gals.  3  qts. ;  how  much 
would  remain  in  the  cask? 

5.  How  much  more  does  a  Johannes  weigh  than  a  doub- 

lom?  (aae^s.) 

6.  If  for  a  horse  worth  £18  10s.  a  man  should  give  a  cow 
worth  £8  7s.  6d.,  and  a  calf  worth  £1  16s.  4d.,  and  the  rest 
in  money ;  how  much  money  would  it  require  ? 

7.  How  much  quicker  could  a  person  travel  from  Lowell 
to  Boston,  in  a  car,  than  in  a  stage,  if  it  should  take  the  car 
1  h.  20  m.  30  sec,  and  the  stage  4  h.  15  m.  and  45  sec.  in 
ths  passage  ? 

8.  How  much  longer  is  the  day  than  the  night,  when  the 
sun  rises  56  minutes  past  4  o'clock,  and  sets  4  minutes  past 
7  o'clock  ? 

9.  What  is  the  difference  of  latitude  between  Boston 
ar  d  Cape  Horn,  Boston  being  42°  28'  north,  and  Cape  Horn 
being  55^  2'  south  latitude  ?  and  how  much  farther  from  the 
Equator  is  Cape  Horn  than  Boston  ? 

10.  What  is  the  third  angle  of  a  triangle,  (4395 14,)  if  the 
three  angles  equal  180°,  the  first  44°  13'  24",  and  the  second, 
79°  46' 38"? 

340.  Model  of  a  Recitation. 

1.  How  much  silver  would  be  required  for  15  spoons,  if  1 
oz.  14  dwt.  12  grs.  be  put  into  each  spoon  ? 

It  would  require  for  15  spoons,  15  times 

lbs.  oz.  dwt.  grs.       as  much  as  for  1  spoon.     15  times  12  grs. 

1  14  12         is  180  grs.,  equal  to  12  grs.,  which  write 

15         in  their  own  column,  and  7  dwts.,  (lOOj) 

— which  add  with  15  times  14  dwts.  making 

2  1  17  12  217  dwts.,  equal  to  17  dwts.,  which  write 
in  their  own  column,  and  10  oz.,  which 
add  with  15  times  1  oz.,  making  25  oz.,  equal  to  1  oz., 
which  write  in  its  own  column,  and  2  lbs.,  which  write  in 
their  own  place, — making  in  all,  2  lbs.  lt)z.  17  dwts.  12  grs., 
which  is  the  answer  required. 


154  arithmetic. 

341.    Exercises  in  multiplying  Compound  Numbers. 

In  like  manner^  solve  and  explain  the  folloioing  problems. 
The  contractions  practised  in  the  muUiplication  and  divisk)n 

of  simple  numbers,  may  be  adopted  here,  whenever  found 

more  convenient. 

1.  If  a  silver  thimble  weigh  12  dwt.  12  grs.,  what  would 
be  the  weight  of  25  thimbles  ? 

2.  If  Lowell  railroad  be  25  m.  5  fur.  30  rods  in  length,  how 
far  would  a  locomotive  run  on  this  road  in  June,  if  it  per- 
form 3  trips  per  day  ? 

3.  How  much  cloth  in  35  pieces,  each  piece  containing 
27  yds.  2  qrs.  3  nails  ? 

4.  How  much  land  in  a  man's  farm  which  is  fenced  into 
9  fields,  each  containing  3  acres,  2  roods,  25  rods  ? 

5.  How  much  gravel  could  a  man  remove  in  18  loads,  at 
1  yd.  25  feet  each  ? 

6.  How  much  would  that  cask  hold,  which  could  be  filled 
with  35  pailfuls,  each  pailful  being  9  qts.  1  pt.  2  gills  ? 

7.  How  many  bushels  of  wheat  in  135  bags,  each  contain- 
ing 2  bush.    3  pks.  ? 

8.  What  would  be  the  weight  of  a  box  of  115  pills,  if  each 
pill  should  weigh  19,  4  grs. 

9.  What  would  be  the  weight  of  15  barrels  of  flour,  at 
196  lbs.  each  ? 

10.  How  much  sterling  money  in  $25,  if  1  dollar  make 
4s.  6d.  ? 

11.  If  a  person  rise  1  h.  20  m.  later  than  he  ought  to  every 
morning  for  12  years,  how  much  time  would  be  thus  wasted  ? 

12.  If  the  sun  appear  to  move  15°  per  hour,  what  is 
its  apparent  motion  in  a  day  of  15.1  hours  in  length  ? 

13.  Multiply  5  oz.  10  dwts.  15  grs.  by  10. 

14.  Multiply  40  m.  3  fur.  25  rods  by  15. 

15.  Multiply-  34  yds.  1  qr.  1  n.  by  64. 

16.  Multiply  57  acres  3  roods  20  rods  by  100. 

17.  Multiply  17  tons  40  ft.  512  in.  by  500. 

18.  Multiply  50  gals.  2  qts.  1  pt.  1  gill  by  25. 

19.  Multiply  £84  15s.  3d.  2  qr.  by  12. 

20.  Multiply  7  tons  1800  lbs.  12  oz.  by  2.5. 

21.  Multiply  5  y.  212  d.  10  h.  15  m.  by  100. 


COMPOUND    NUMBERS.  155 


S'US*   Model  of  a  Recitation. 

1.  If  2  lb.  1  oz.  17  dwt.  12  grs.  of  silver  be  put  into 
It  spoons,  what  would  be  the  weight  of  each  spoon  ? 

lbs.  oz.  dwts.  grs.  Each  spoon  would  weigh  -^  as 

2     1    17    12  much  as  15  spoons.     As  15  is  not 

12  contained  in  the  2  lbs.  reduce  them 

—  to  ounces,  making  with  the  1  oz. 

It) 25  25  ounces,  which  divided  by    15 

— -  gives  1  oz.    for  the  quotient,  and 

1  oz.  10  10  oz.   remainder,  which  reduced 

20  to  pennyweights,  make  with  the  17 

*  dwts.  217  dwts.,  which  divided  by 

15)  217  15  gives  14  dwts.  for  the  quotient, 

—  and  7  dwts.  remainder,  which  re- 

14  dwts.  7  duced  to  grains,  make  with  the  12 

24  grs.  180  grs.,  which  divided  by  15 

gives  12  grains  for  the  quotient, 

15)180  — making  in  all,   1  oz.    14  dwts. 

12  grs.,  which  is  the  answer  re- 

1  oz.  14  dwts.  12  grs.     quired. 

3^t3.    Exercises  in  dividing  Compound  Numbers. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  What  would  be  the  weight  of  1  dollar,  the  weight  of  8 
dollars  being  6  oz.  18  dwts.  16  grs.  ? 

2.  How  far  does  that  boy  live  from  his  school-house,  who 
has  to  travel  170  m.  2  fur.  in  attending  school  twice  a  day, 
for  60  days  ? 

3.  How  long  is  that  room  which  requires  27  yds.  2  qrs.  of 
carpeting  cut  into  5  pieces,  to  carpet  it  ? 

4.  If  135  acres  be  fenced  off  into  16  equal  lots,  what  would 
be  the  size  of  each  lot  ? 

5.  If  a  man  team  to  market  25  cords,  5  cord-feet  of  wood, 
at  20  loads ;  how  much  would  that  be  a  load  ? 

6.  What  is  the  contents  of  each  of  such  bottles  that  160 
of  them  could  be  filled  from  a  cask  holding  115  gallons  ? 


156  i^RITHMETIC. 

7.  How  many  apples  in  a  barrel,  if  101  bush.  1  pk.  make 
45  barrels  ? 

9.  What  would  be  the  weight  of  a  dose  of  medicine,  if 
4  g,  5  5,  2  9,  12  grs.  be  taken  at  12  doses  ? 

9.  If  33  lbs.  of  steel  be  put  into  256  axes,  how  much 
would. that  be  apiece  ? 

10.  If  147  bushels  cost  47£.  12s.  5d.,  what  does  it  cost 
per  bushel  ? 

11.  If  a  teacher  devote  5  hrs.  30  min.  per  day  to  50 
scholars,  how  much  would  that  be  for  each  scholar? 

12.  What  would  be  the  daily  motion  of  the  moon,  if  it 
complete  a  revolution  in  29|  days  ? 

S4J:.   Model  of  a  Recitation. 

How  many  rods  round  a  pasture,  measuring  on  the  first 
side  ^  of  a  mile,  on  the  second  1.17  furlongs,  on  the  third 
2^  furlongs,  and  on  the  fourth  1  furlong  18.8  rods, 
^^m.  =jJL>^|;i^  rods  =110   rods. 

1.17  fur.  =1.17  X  40  rods        =   46.8    " 

2/^  fur.  =2/^X40  rods  =:   86 

1  fur.  18.8  rods  =  40  rods +  18.8  rods  =   58.8    " 


301.6  rods. 

These  quantities  being  expressed  in  different  ways,  mvj^t^ 
before  they  can  be  added,  he  reduced  to  numbers  of  the  same, 
kind. 

Multiply  ^^  m.  by  8  to  reduce  it  to  furlongs,  and  that  pro- 
duct, also,  1.17  furlongs,  2^g-  furlongs,  and  1  furlong,  by  40  to 
reduce  them  to  rods ;  these  several  quantities,  being  made 
alike  and  added,  make  301.6  rods,  which  is  the  answer 
required. 

S4l^*     Exercises   in    the    use    of    Numbers    variously 

EXPRESSED. 

In  like  mannei ,  solve  and  explain  the  following  problems, 

1.  How  much  is  -f^  of  a  week  and  J  of  a  day  ? 

2.  What  is  the  difference  between  two  fields,  one  of  which 
measures  2|-  acres,  the  other  1  acre  2|-  roods  ? 

3.  How  much  cloth  in  three  remnants,  tbe  first  measuring 
2.4  yards,  the  second  1  yd.  3  qrs.,  and  the  third  f  of  a  yard  ^ 

4.  How  many  cubic  inches  in  4  bushels,  1.375  bushels 
and  ^  of  a  peck  ? 


COMPOUND   NUMBERS.  157 

5.  If  2|  pecks  be  taken  from  a  bag  holding  2.75  bushels ; 
ho  .y  much  would  be  left  ? 

3.  How  much  water  in  a  pail  measuring  10.75  quarts,  but 
wanting  -^^  of  a  gallon  of  being  full  ? 

7.  How  much  silver  in  a  large  and  small  spoon,  sugar- 
tor  gs,  and  butter-knife,  weighing  severally,  2.9  oz.,  1  oz. 
12|.  dwt..  If  oz.  and  18.5  dwt.? 

B.  If  an  apothecary  should  mix  a  medicine  at  a  cost  of 
$A3  per  ounce,  and  should  sell  it  at  $.48  per  ounce  avoirdu- 
pos;  how  much  would  he  gain  in  selling  10  lbs.? 

).  Add  -^  of  a  ton,  16^^  lbs.,  ^\  of  a  ton,  and  .83  of 
a  tDn  together. 

10.  Subtract  |  of  a  shilling  from  £1.25. 

11.  What  is  the  difference  between  52  wks.  1  d.  6hrs.  and 
365.25  days?  - 

12.  If  two  ships  sail  from  the  same  point,  one  north  IB/^ 
degrees,  the  other  south  25°  33^' ;  what  then  would  be  the 
lat  tude  between  them  ? 

S416.  Model  of  a  Recitation. 

How  many  years,  months  and  days  from  the  first  resistance 
with  arms  in  the  American  revolution,  April  19th,  1775,  to 
the  declaration  of  Independence,  July  4th,  1776? 

To  answer  this  question  you  must  subtract  the  time 
between  the  Christian  era  and  the  earlier 

1775  ^6     3^        date,  which  is  1774  years,  3  months  and 

1774  3  18  -^^   days,   from   the    time    between   the 
Christian  era  and  the  later  date,  which  is 

1     2  15  1775   years,  6  months  and  3  days,  the 

hours,  &c.,  being  disregarded. 

Reduce  1  of  the  6  months  to  days,  making,  (20^5 )  with 

the  3  days,  33  days,  from  which  18  being  subtracted,  15  days 

remain ;  3  months  from  the  other  5  months  leave  2  months  ; 

and  4  years  from  5  years   leave  1  year,  making,  in  all,  1 

year,  2  months  and  15  days,  which  is  the  answer  required. 

But  a  more  convenient  way  of  obtaining  the  same  result 

is,  instead  of  writing  the  number  of  years, 

1776  7     4  months  and  days,  to  write  the  order  of 

1775  4  19  the    year,  month  and    day,  that  is,  the 
dates  themselves.    Thus,  from  the  1776th 

1     2  15  year,  7th  month  and  4tn  day,  subtract  the 

1775th  year,  4th  month  and  19th  day ; 
14 


j5S  arithmetic. 

precisely  as  so  many  years,  months  and  days.  This  increases 
each  number  in  each  denomination  by  the  same  quantity,  1, 
and,  consequently,  does  not  affect  the  difference. 

24:T«    Exercises   in   finding   the    Difference   of   Time 
BETWEEN  Dates. 

In  like  manner,  solve  and  explain  the  following  problems, 

1.  How  long  from  the  time  that  Washington  entered  up/^n 
the  command  of  the  American  army  at  Cambridge,  July  '2d, 
1775,  till  the  disbanding  of  the  army  at  West  Point,  Novem- 
ber 3d,  1783  ? 

2.  How  long  was  General  Harrison's  victory  at  Tippecanoe, 
November  7th,  1811,  before  General  Jackson's  victory  at 
New  Orleans,  January  Sth,  1815  ? 

3.  How  long  from  the  date  of  a  note.  May  10th,  1835,  till 
its  payment,  June  5th,  1840  ? 

4.  How  old  is  that  man,  June  27th,  1840,  who  was  born 
March  23d,  1807  ? 

5.  What  is  the  date  of  that  note,  which  was  paid  Decem- 
ber 31st,  1839,  2  y.  3  m.  11  d.  after  its  date? 

6.'  When  was  that  note  paid,  which  was  dated  August  4th, 
1836,  and  paid  3  y.  3  m.  30  d.  after  date  ? 

7.  Yiow  much  older  is  Lizzie,  born  Sept.  11th,  1843,  than 
Mary,  born  April  15ih,  1846  ? 

8.  How  long  was  John  absent,  having  left  town  July  1st, 
1839,  and  returned  August  25th,  1840? 

248.    Model  of  a  Recitation. 

What  cost  4  bu<  'a.  3  pks.  1  gal.  of  wheat,  at  5s.  6d.  per 
bushel  ? 

The  whole  cost  will  be  the  product  of  the  price  of  one 

bushel  by  the  number  of  bushels; 

4.875  bushels.         but,  before    they  can  be  multipliea 

.275  £.  together,  they   must  be  reduced  to 

^  simple   numbers ;  4   bush.    3   pks. 

24375  1  gal.  reduced  to  bushels  is  4.875 

34125  bushels,  (^SS,)  and  5s.  6d.  reduced 

9750  to  pounds  is  £.275  (230.)     Now, 

since  1  bushel  costs  £.275,  the  whole 


1.340625£=  cost  will  be  .275  as  many  pounds 

1£.  6s.9|d.  as    bushels,  which    is    £1.340625 

equal  (231)  to  1£.  6s.  9|d. 


COMPOUND    NUMBERS.  159 

S^9.   Exercises  in  the   Reduction  of  Compound  Num- 
bers FOR  Multiplication. 

In  like  manner^  solve  and  explain  the  following  problems. 

1.  What  is  the  value  of  15  acres,  2  roods,  20  rods,  $62.25 
being  the  cost  of  each  acre  ? 

2.  What  would  12  miles,  3  furlongs,  32  rods  of  road  cost, 
at  175£.  10s.  6d.  per  mile? 

3.  Goliah,  measuring   6 J    cubits    of  1    ft.  7.168   in.   in 
height,  was  how  tall  in  feet  and  inches  ? 

4.  What  is  the  cost  of  5  yds.  1  qr.,  2  nls.  of  broadcloth,  at 
$5.50  per  yard ? 

5.  What  is  the  value  of  2^  tons,  1|  tons,  and  2|  tons  of 
bay,  at  $12.25  per  ton? 

6.  What  will  4f  tons  of  iron  come  to,  at  20£.  15s.  6d.  per 
tm? 

7.  What  will  8f  hogsheads  of  molasses,  at  63   gallons 
each,  come  to,  at  2s.  6d.  per  gallon  ? 

8.  At  5s.  per  bushel,  what  will  4  bush.  2  pks.  1  qt,  of  corn 
come  to  ? 

9.  Bought  a  silver  cup,  weighing  9  oz.  4  dwt.  16  grs.,  at 
Cs.  8d.  per  ounce,  what  was  the  whole  cost? 

S^SO.   Model  of  a  Recitation. 

How  many  square  feet  in  a  square,  measuring  16  ft.  6 
in.  on  each  side  ? 

16.5 

16.5  Since  the  contents  is  the  product  of  the 
length  by  the  breadth,  (lOS,)  and  16  feet 

825  6  inches  being  16.5   feet,  the  contents  will 

990  be  16.5  X  16.5  =  272.25  square  feet,  which 

165  is  the  answer  required,  (4395  ^^•) 


272.25  feet. 

3S1*   Exercises  in  the  Mensuration  of  Surfaces  and 
Solids. 

In  like  manner,  solve  and  explain  the  following  prohleTns, 

1.  How  many  square  inches  (105)  on  the  page  of  a  book 
8  inches  long  and  5  inches  wide  ? 

2.  How  many  square   yards  in  a  square,  measuring  5 J 
yards  on  each  side  ? 


160  ^  AHITHMETIC. 

3.  How  many  feet  in  a  floor  which  is  16|  feet  long  and  15 
feet  wide  ? 

4.  How  many  square  yards  will  carpet  a  floor  which  is  5 
yds.  1  ft.  6  in.  long,  and  5  yards  wide  ? 

5.  How  many  rods  in  a  garden  5  rods,  2|  yards  long,  and 
4.5  rods  wide  ? 

6.  How  much  land  in  a  field  26  rods,  11  feet  long,  and  6 
rods  wide  ? 

7.  How  many  feet  in  a  board  17  ft.  9  in.  long,  and  1  ft. 
6  in.  wide  ? 

8.  If  from  a  square  stick  of  timber  1  foot  wide  and  1  foot 
thick,  you  saw  off  a  piece  1  foot  long,  that  block  would 
contain  exactly  1  cubic  foot;  how  many  cubic  feet  in  such  a 
stick  of  timber  16  feet  long  ? 

9.  How  many  boards  1  inch  thick  could  be  made  of  that 
stick,  allowing  no  waste  in  sawing  ? 

10.  How  many  cubic  inches  in  one  of  the  boards  ? 

11.  How  many  cubic  inches  in  all  of  the  boards  ? 

12.  How  many  cubic  inches  in  a  stick  of  timber  1  foot 
wide  and  thick,  and  16  feet  long  ? 

13.  How  many  feet  in  a  stick  of  timber  24  feet  long,  1.8 
feet  wide,  and  1.5  feet  thick? 

14.  How  many  feet  in  2  sticks  of  timber,  each  36  feet 
long,  2  ft.  6  in.  wide,  and  2  ft.  3  in.  thick  ? 

15.  How  many  feet  in  a  load  of  wood  8  ft.  long,  3  ft.  6  in. 
wide,  and  3  ft.  9  in.  high  ? 

16.  How  many  feet  in  a  load  of  gravel  7  ft.  6  in.  long, 
4  ft.  3  in.  wide,  and  2  ft.  3  in.  high? 

17.  How  many  yards  of  gravel  must  be  removed  to  make 
a  cellar  2.5  yards  deep,  6  yards  long,  and  5.6  yards  wide  ? 

IS.  How  many  yards  of  stone  work  in  a  wall  38|  yards 
long,  4  ft.  6  in.  high,  and  .8  of  a  yard  thick? 

19.  How  many  feet  in  a  room  17  ft.  6  in.  long,  15  ft.  3 
in.  wide,  and  10  ft.  9  in.  high  ? 

20.  How  many  cord-feet  in  a  load  of  wood  8  feet  long,  4 
feet  wide,  and  4  feet  high  ? 

21.  How  many  cords  of  wood  in  a  pile  32  feet  long,  4  feet 
wide,  and  7  feet  high  ? 

333.   Illustration  of  the  mode  of  Abridging  the  Pro- 
cess OF  solving  Problems. 

1.  What  is  the  value  of  a  pile  of  wood  64  feet  long,  4  feet 
wide,  and  6  ft.  6  in.  high,  at  $5.25  per  cord  ? 


COMIOUNP    NUMBERS.  161 

In  questions  like  this,  involving  both  multiplication  and 
division,  it  will  be  most  convenient,  and  will  generally  much 
abridge  the  process,  to  express  all  the  operations  before  per- 
forming  any  of  them.     Thus;    the   length  64  feet  muhi- 

plied  by  the 
64  X  4  X  6.5  H-  16  ~  8  X  5.25  =  $68.25.         breadth  4  ft. 

will  give  the 
s:iuare  contents  of  the  base,  which  multiplied  by  the  height 
C  ft.  6  in.,  or  Q.5  feet,  will  give  the  cubic  contents  (190) 
ii  feet;  this  divided  by  16  will  give  the  contents  in 
cord-feet,  and  this  quotient  divided  by  8  will  give  the  con- 
t3nts  in  cords,  which  multiplied  by  the  price  of  1  cord,  will 
give  the  whole  value,  or  the  answer  required. 

The  whole  process  being  thus  expressed  and  explained, 
jerform  the  operations  indicated  by  the  signs,  in  such  order  as 
v/ill  require  the  fewest  figures;  thus,  divide  the  64 by  the  16 ; 
tbe  quotient,  4,  multiply  by  the  4;  the  product,  16,  divide 
hy  the  8;  multiply  Q.3  by  the  quotient,  2,  and  multiply 
$5.25  by  that  product,  13,  making  $68.25,  which  is  the 
answer  required. 

Perhaps  it  will  be  more  convenient  still,  to  express  the 
process  in  a  fractional  form,  (865)  by  making  the  divisors 
factors  of  the  denominator,  and  then  to  reduce  the  fraction 

to  its  lowest  terms. 
^      2  (121.)  Thus,  the  16 

^^y^^  64  in  the  numera- 

tor, may  be  canceled 
from  both  terms  (ISl) ;  and  4,  the  other  factor  of  64,  multi- 
plied into  the  other  4  of  the  numerator,  makes  16,  of  which 
the  8  in  the  denominator  is  a  factor;  consequently,  8  may  be 
canceled  from  both  terms ;  and  2,  the  other  factor  of  16,  mul- 
tiplied into  Q,5  makes  13,  which  multiplied  into  $5.25  makes 
$68.25,  as  before. 

S^3*   Exercises   in   solving    Problems    involving   both 
Multiplication  and  Division. 

In  like  manner,  solve  and  explain  the  folloioing  problems. 
1.    How  many  cords  of  wood  in  a  pile  40  feet  long,  4  feet 
wide,  and  9  ft.  3  in.  high  ? 

14* 


162  ARITHMETIC. 

2.  What  is  the  value  of  a  load  of  wood,  measuring  S  feet 
in  length,  4  ft.  6  in.  in  width,  and  5  ft.  3  in.  in  height,  at 
$8  per  cord  ? 

3.  What  is  the  value  of  a  stick  of  timber,  measuring  50 
feet  in  length,  2  ft.  6  in.  in  width  and  thickness,  at  $4  per 
ton? 

4.  What  would  be  the  cost  of  digging  a  cellar  19  ft.  6 
in.  long,  15  feet  wide,  and  10  ft.  6  in.  deep,  at  $.25  per 
yard? 

5.  How  many  acres  in  a  pasture  36  rds.  8.25  ft.  long,  and 
30  rods  wide  ? 

6.  How  many  yards  in  a  floor  28  ft.  9  in.  long,  and  22 
ft.  4  in.  wide  ? 

7.  How  many  yards,  in  length,  of  carpeting,  which  is  4 
ft.  6  in.  wide,  will  cover  a  floor  17  feet  long,  and  15  ft.  6 
in.  wide  ? 

8.  How  many  days  would  Samuel  have  to  go  to  school, 
twice  per  day,  to  travel  1000  miles,  if  he  live  5  furlongs 
from  school  ? 

9.  How  many  times  would  a  wagon  wheel,  13  ft.  9  in. 
in  circumference,  revolve  in  running  25  miles,  6  furlongs  ? 

10.  How  many  times  could  a  coal-basket,  holding  1 
bush.  1  gal.  2  qts.,  be  filled  from  a  coal-cart,  containing  65 
bush.  1  pk.  2  qts.  ? 

11.  What  would  a.hogshea4  of  cider,  containing  62  gals. 
2  qts.  come  to,  at  $.04  per  bottle  of  1  pt.  2  gills  ? 

12.  What  is  the  value  of  a  lot  of  spoons,  weighing  9  lbs. 
10  oz.  4  dwt.,  each  spoon  weighing  16  dwt.  10  grs.,  and 
worth  $1.25? 

13.  How  many  loads  of  hay,  each  weighing  1750  lbs.,  in 
a  stack,  weighing  16  tons,  875  lbs  ? 

14.  How  many  pills  may  be  made  of  a  mixture  of  lOg 
43,  each  weighing  13,10  grs.  ? 

15.  In  67£.  lis.  7d.,  how  many  crowns,  at  6s.  7d.  each  ? 

16.  How  many  yards  of  English  cassimere,  at  12s.  8d. 
per  yard,  may  be  bought  for  395£.  4s.  sterling  ? 

17.  What  part  of  £1,  or  20s.  is  15s.  ? 

18.  If  a  yard  of  broadcloth  cost  17s.  6d. ;  what  part  of  a 
yard  might  be  bought  for  13s.  6d.  ? 

19.  What  part  of  £1  15s.  9d.  is  15s.  9d.? 


COMPOUND   NUMBERS.  163 

9«S4.   Model  of  a  Recitation 

Reduce  £64  17s.  6d.  of  sterling  money,  and  of  Canada, 
!^"ew  England,  New  York,  Pennsylvania,  and  Georgia,  cur- 
rencies to  Federal  money;  and  the  results  back  again  as 
bifore. 

i*fi4.  R7^  Reduce     17s.     6d.     to 

^^  the   decimal  of  a  pound, 

(SSOj)     and,    since    in 

9)2595.000  ^'^'^""^  7"^y   \^J 

' one    pound   equals    $4|, 


$288.33^ 


there  will  be  4|,  or  ^ 

q^  times    as    many    dollars 

as   pounds,  which    gives 


40  \  2595  00  S288.33^. 

'  Since  one  dollar  equals 

£64.875  =  £64.  17s.  6cl.        ^ '  '^^'^  f'"  ^,  f  ^' 

many  pounds  as  dollars ; 

which  gives  £64.875. 

£64.875  Since,  in  Canada  currency,  (SOS,)  one 

4  pound  equals   $4,  there  will  be  4  times 

as  many  dollars  as  pounds ;  which  gives 

4 )  $259.5000  $259.50. 

Since    1    dollar  equals  £J,  there  will 

£64.875.        be  J  as  many  pounds  as  dollars ;  which 
gives  £64.875. 

Since,  in  N.  England  currency,  (SOS^) 

3 )  £64.875         one  pound  equals  $3  J ,  there  will  be  3 J ,  or 
^^  times  as  many  dollars  as  pounds;  divide 

$216.25         by  3  to  obtain  |,  and  remove  the  point 
.3         one  place  farther  to  the  right  to  obtain 

J# ;  which  gives  $216.25. 

£64.875.  Since  $1  =?  £.3,  there  will   be  .3  as 

many  pounds  as  dollars,  or  £64.875. 
Since,  in  New  York  currency,  (SM>tl,) 

4)  £64.875        £1  =  $2|,    there    will    be   21,    or    J^ 

times  as  many  dollars  as  pounds  ;    which 

$162.18f       gives  $162.18|. 

.4  Since    $1  =  £.4,' there  will  be  .4  as 

many  pounds    as   dollars;    which   gives 

£64.875.       £64.875. 


164  ARITHMETIC. 

£64.875 

8 

Since,     in     Pennsylvania     currency, 

3  )  519.000  (SO^,)  £1  =  $2|,  there  will  be  2|,  or 

f    times    as    many    dollars    as    pounds ; 

$173.  which  gives  $173. 

3.  Since   SI  ==  £§,  there  will   be   f   as 

many  pounds  as   dollars;    which   gives 

8)519.  £64.875. 


£64.875. 

£64.875 
30 


Smce,    m  Georgia  currency,   (90li.) 

7 )  1946.250  £1  =  ^f ,  there  will  be  42,  or  ^  times 

as  many  dollars  as  pounds ;  which  gives 

$278,034-  $27S.03f 

7  Since  $1  =  £-^^,  there  will  be  -/^  as 

many  pounds    as  dollars ;    which   gives 

30 )  1946.25  £64.875. 


£64.875. 


3^^.    Exercises  in- the  Reduction  of  Currencies. 

In  like  wanner,  solve  and  explain  the  following  proh' 
lems. 

1.  One  dollar  is  what  part  of  a  pound  in  sterling  money, 
and  in  Canada,  New  England,  New  York,  Pennsylvania, 
and  Georgia  currencies  ? 

2.  What  part  of  one  dollar  is  one  pound  of  sterling  money, 
and  of  each  of  the  currencies  ? 

3.  How  many  dollars  in  £1  of  sterling  money,  and  of 
each  of  the  currencies  ? 

4.  Reduce  £25  10s.  from  sterling  to  federal  money. 

5.  Reduce  $25,375  to  sterling  money. 

6.  How  much  Federal  money  would  pay  for  a  farm,  in 
Canada,  worth  £500. 

7.  If  a  Canadian  lumber  merchant  sell,  in  New  Orleans 
lumber  that  cost  him  £875  for  $3750,  whether,  and  how 
much,  would  he  gain,  or  lose  ? 


COMPOUND    NUMBERS.  166 

8.  If  a  farm,  in  Cambridge,  Massachusetts,  which,  in  1740, 
cost  £125  7s.  6d.,  be  now  worth  $3000,  how  much  -has  it 
iiicreased  in  value  ? 

9.  How  many  dollars  should  a  New  York  merchant 
receive  for  125  yards  of  flannel,  at  3s.  6d.  per  yard? 

10.  If  a  wholesale  dealer,  in  Philadelphia,  receive  $1500 
fcr  a  quantity  of  cloth,  at  3s.  9d.  per  yard ;  how  many  yards 
ir .  the  quantity  ? 

11.  If  a  Georgia  planter  sell  his  wheat  at  3s.  6d.  per 
bushel,  and  receive  $387.50 ;  how  many  bushels  would  he 
sell? 

12.  Where  can  the  same  kind  of  penknives  be  bought  the 
clieapest,  if  2s.  3d.  apiece  be  the  price  ? 

13.  How  much  Federal  money  would  a  horse  cost  in  each 
o:'  the  several  currencies ;  if  the  price  be  £20  18s.  ? 

14.  Eeduce  $.25  to  each  of  the  several  currencies. 

15.  Reduce  Is.  6d.  of  the  several  currencies  to  Federal 
ironey. 

Note.  Whenever  any  of  the  denominations  of  English 
rrioney  occur  in  the  following  pages,  they  will  be  in  New 
E  ngland  currency,  unless  otherwise  specified. 

9!56«  Model  of  a  Recitation. 

1.  A  merchant  sold  3545  yards  of  cotton  cloth,  at  9d.  per 
yard ;  what  was  the  amount  of  it  in  Federal  money  ? 

Since  the  price  of  1  yard  was  |^  of  a  dollar,  (SOOj)  the 

amount  of  the  whole  quantity  must  have 

8)3545  been  -j-  as  many  dollars   as  there  were 

yards  ;  therefore,  divide  (93)  the  number 

$443,125.         of  yards  by  8  to  ascertain  how  ma^y  dol- 
lars there  were  in  the  amount. 

2.  A  merchant  paid  $35.31  J  for  cotton  cloth,  at  4Jd.  per 
yard ;  how  many  yards  did  he  purchase  ? 

Since  the  price  of  1  yard  was  -^^  of  a  dollar,  (SO65)  he 
dUQCQii  must  have  purchased  16  times  as  many 

*      tfi  y^^^^  ^^  ^^  P^^^  ^^^^^^^  (*^1) '  therefore, 

multiply  the  number  of  dollars  that  he  paid 


t^nt-         1  by  16  to  ascertain   how  many  yards  he 


purchased. 


166  arithmetic. 

3^7.   Observation. 

Hence,  observe,  that,,(^i5^^)  in  the  following  questions, 
and  lohenever  the  multiplier,  or  divisor,  is  such  that  it  can  be 
reduced  to  an  aliquot  part  of  a  dollar,  the  process  nmy  he  much 
abridged,  by  using  the  aliquot  part. 

958.    Exercises  in  the  Use  of  Aliquot  Parts. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  What  cost  4S72  oranges,  at  3d.  apiece  ? 

2.  How  many  pounds  of  sugar,  at  6d.  per  pound,  may  be 
purchased  with  $1^.58^  ? 

3.  How  many  dollars  would  it  take  to  pay  for  144  yards 
of  calico,  at  Is.  per  yard  ? 

4.  How  many  times  is  Is.  3d.  contained  in  $5  ? 

5.  What  is  the  value  of  1728  bushels  of  apples,  at  Is.  6d.  a 
bushel  ? 

6.  How  many  bushels  of  potatoes  would  it  take  to  come  to 
$24.66|,  at  $.33J  per  bushel  ? 

7.  What  would  be  the  cost  of  24  yards  of  muslin,  at  2s.  3d 
per  yard  ? 

8.  Bought  36  yards  of  bombazine,  at  2s.  6d.  per  yard. 
What  was  the  bill  ? 

9.  How  many  pairs  of  half-hose,  at  $.45f  a  pair,  may  be 
bought  for  $11  ? 

10.  What  would  be  the  cost  of  40  yards  of  linen,  at  3s.  per 
yard  ? 

11.  Sold  children's  shoes,  at  3s.  3d.  a  pair,  to  the  amount 
of  $54.16| ;  how  many  pairs  were  sold  ? 

12.  How  many  palm-leaf  hats,  at  3s.  6d.  apiece,  may  be 
bought  with  $14.58J  ? 

13.  What  may  I  receive  for  576  lbs.  of  wool,  at  $.62J  a 
pound  ? 

14.  How  much  flannel,  at  4s.  a  yard,  may  be  bought  with 
$4.66§  ? 

15.  If  the  expense  of  cultivating  an  acre  of  corn  be  $20, 
what  would  be  the  profits  from  a  field  of  12  acres,  each  yield- 
ing 50  bushels,  worth  4s.  6d.  per  bushel  ? 

16.  If  a  farmer  sell  his  rye  at  $.83J  per  bushel,  and 
receive  $95  for  it,  how  many  bushels  would  he  sell  ? 

17.  How  many  days,  at  5s.  3d.  per  day,  must  a  man  work 
to  earn  $63.87^  ? 


COMPOUND    NUMBERS.  167 

18.    If  a  merchant  sell  4  dozen  pairs  of  gloves,  at  5s.  6d.  a 
pair,  what  would  he  receive  for  them  ? 

3il9.   Model  of  a  Recitation. 

At  7s.  6d.  a  bushel,  what  would  100  bushels  of  wheat  come 
to? 

Since  7s.  6d.,  the 

4.)  $100  =  cost  at  $1.      per  bushel.         ?ji^\^Ui^'' wt^: 
J5  =  cost  at  $_^  per  bushel.         ^^^f^^^^^ 

$125  =  cost  at  $1.25,  or  7s.  6d.  per     -^bertf  ddlS 

bushel.  •     ^u  ...    1 

m  the   cost   at  1 

dollar  per  bushel, 
arid  J  of  the  number  of  buShels  would  be  the  number  of  dol- 
lars in  the  cost,  at  J  of  a  dollar  per  bushel ;  therefore,  write 
the  number  of  bushels,  and  to  it  add  J  of  itself,  and  the  sum 
will  be  equal  to  the  whole  cost  in  dollars. 

S(&0.   Exercises  in  multiplying  by  Units  and  Aliquot 
•Parts. 

In  like  manner i  solve  and  explain  the  following  problems, 

1.  How  much  would  12  pairs  of  ladies'  shoes  come  to,  at 
6s.  6d.  a  pair? 

2.  At  6s.  9d.  a  pair  for  silk  hose,  what  would  be  the  price 
per  dozen  ? 

3.  What  would  be  the  cost  of  54  gallons  of  oil,  at  7  shil- 
lings per  gallon  ? 

4.  How  much  would  5J   pounds  of  green  tea,  at  8s.  a 
pound,  amount  to  ? 

5.  Bought  2.875  yards  of  satinet,  at  8s.  3d.  per  yard  ; 
what  was  the  cost  ? 

6.  How  much  would  11  pitchforks,  at  9s.  apiece,  amount 
to? 

7.  Multiply  1840  by  If 

8.  What  would  12  weeks'  work  come  to,  at  10s.  6d.  per 
day,  Sundays  excepted  ? 

9.  How  much  would  2^  yards  of  cassimere  cost,  at  lis. 
3d.  per  yard  ? 

10.  What  would  be  the  cost  of  52  weeks'  board,  at  15s. 
per  week  ? 


168  ARITHMETIC. 

11.  Paid  for  18  weeks'  board,  at  13s.  6d.  per  week  ;  how 
much  was  the  bill  ? 

^61.   Illustration  of  the  Principle  of  reducing  Frac- 
tions BY  Inspection. 

Since  100  cents  make  the  unit,  1  dollar,  any  number  of 
cents  are  so  many  hundredths  of  a  unit ;  thus,  12|  cents  is 
S.125.  But  12 J  cents  is  a  ninepence,  or  ^  of  a  dollar 
(S06) ;  therefore,  the  decimal  for  any  number  of  eighths  will 
be  the  number  of  cents  in  so  many  ninepences  ;  and,  for  any 
number  of  twenty-fourths,  sixteenths,  twelfths,  and  sixths,  the 
decimal,  in  hundredths,  will  be  the  number  of  cents  in  so 
many  threepences,  fourpence-halfpennies,  sixpences,  and  shil- 
lings, respectively. 

Also,  any  decimal,  corresponding  with  the  number  of  cents 
in  any  such  part  of  a  dollar,  may  be  reduced  to  a  common 
fraction,  by  writing  the  part  instead  of  the  decimal. 

363.   Model  of  a  Recitation. 

1.  Reduce  -f-^  to  an  equivalent  decimal. 

Fourpence-halfpennies  being  sixteenths 

5   *^19^         ^^  ^  dollar,  the  decimal  for  -^^  will  be  the 

^^        *  number  of  cents  in  5  fourpence-halfpen- 

nies, which   is   31J  ;    consequently,  the 
decimal  required  is  .3125. 

2.  Reduce  .4183  to  an  equivalent  common  fraction. 

The  hundredths,  in  this  decimal,  being 

A-icjf^ 5^         the  same  as  the  number  of  cents  in  2s.  6d. 

T2         =.5  sixpences,  and  sixpences  being  twelfths 
(SOG)  of  a  dollar,  this  decimal  is  equiva- 
lent to  -^,  which  is  the  answer  required. 

363*   Exercises  in  reducing  Fractions  by  Inspection. 

In  like  manner ,  solve  and  explain  the  following  problems, 

1.  Reduce  ^V  to  a  decimal. 

2.  Reduce  .2083  to  a  common  fraction. 

3.  What  are  the  decimals  equivalent  to  ^,  ^^,  and  ^  ? 

4.  What  are  the  common  fractions  equivalent  to  .7083, 
.7916,  and  .9583  ? 


r 


PROPOETION.  169 


5.  Reduce  -j^  to  a  decimal. 

6.  Reduce  .1875  to  a  common  fraction. 

7.  What  are  the  decimals  equivalent  to  ^^,  f^,  and  -f-J  ? 

8.  What  are  the  common  fractions  equivalent  to  .8125  and 
.^•375  ? 

9.  Reduce  -j^  to  a  decimal. 

10.  Reduce  .583  to  a  common  fraction. 

11.  What  is  the  decimal  equivalent  to  -J^  ? 

12.  What  are  the  common  fractions  equivalent  to  .375, 
.( 25,  and  .875  ? 

13.  What  are  the  decimals  equivalent  to  -J  and  |-  ? 

14.  What  are  the  decimals  equivalent  to  ^  and  |  ? 

15.  Reduce  .33J  and  .66  to  common  fractions. 

16.  Reduce  5  inches  to  the  decimal  of  a  foot. 

17.  Reduce  5  ounces  to  the  decimal  of  a  pound  avoirdu- 
pois. 

18.  How  many  ounces  in  .583  of  a  pound  Troy  ? 

19.  Reduce  7  grains  to  the  decimal  of  a  pennyweight. 

20.  How  many  furlongs  in  .625  of  a  iriile  ? 


IX.    PROPORTION. 

3<64..  Illustration  of  Ratios. 

When  two  quantities  of  the  same  kind  are  compared  with 
regard  to  their  relative  value,  one  of  them  will  be  less  than, 
equal  to,  or  greater  than,  the  other ;  and  will  contain  the 
other  less  than  once,  exactly  once,  or  more  than  once. 

•The  Ratio  of  one  quantity  to  another  of  the  same  kind,  is 
the  quotient  resulting  from  the  division  of  the  latter  by  the 
former ;  the  division  being  expressed  in  a  fractional  form,  or, 
more  frequently,  with  the  dividend  following  the  divisor  with 
this  sign  (  !  )  between. 

:  is  the  sign  for  the  ratio  of  two  quantities.  It  indicates 
the  ratio  of  the  antecedent,  or  the  quantity  preceding  the  sign, 
to  the  consequent,  or  the  quantity  which  follows  the  sign. 

The  ratio  of  two  numbers  shows  what  part  the  dividend  is 

of  the  divisor.     Thus,  in  comparing  7  dollars  with  12  dollars, 

4  fathoms  with  8  yards,  and  11  with  3,  we  find  that  7  dollars 

is  -j^  of  12  dollars,  and  contains  12  dollars  -^  of  one.  time, 

15 


170  AKITHMETIC. 

and  that  their  ratio  is  12  :  7  ;  that  4  fathoms,  being  equal  to 
8  yards,  is  f  of  8  yards,  and  contains  8  yards  f  of  one  time, 
or  exactly  once,  and  that  their  ratio  is  8  :  8,  which  is  called 
the  ratio  of  equality^  since  the  two  terms  of  the  ratio  are 
equal ;  and,  finally,  that  11  is  -V-  of  3,  and  contains  3  -y-  of 
one  time,  or  3|^times,  and  that  their  ratio  is  3  :  11. 

Hence,  (87^)  to  ask  what  part  of  12  dollars  is  7  dollars, 
is  the  same  as  to  ask  what  is  the  ratio  of  12  dollars  to  7  dol- 
lars, or  of  12  to  7,  since  -^^  is  the  part  of  12  that  7  is,  and 
also  the  ratio  of  12  to  7. 

Consequently,  any  fraction  is  the  ratio  of  its  denominator 
to  its  numerator  ;  and  in  writing  a  ratio  fractionally,  the  first 
number  is  made  the  denominator,  or  divisor,  and  the  second 
the  numerator,  or  dividend.  Thus,  12  :  7  is  read,  the  ratio 
of  12  to  7,  and  is  the  same  as  -pV* 

SOil.   Exercises  in  finding  the  Ratios  of  Numbers. 

In  like  manner^  solve  and  explain  the  following  prohlems. 

1.  What  part  of  7  is  3,  and  what  is  the  ratio  of  7  to  3  ? 

2.  What  part  of  5  is  12,  (ST.)  and  what  is  the  ratio  of  5 
to  12  ? 

3.  V/hat  part  of  8  is  f,  (ll^,)  and  what  is  the  ratio  of  8 
tot? 

4.  What  part  of  10  is  31,  (US,)  and  what  is  10  :  3J  ? 

5.  What  part  of  |  is  4,  (ISS,)  and  what  is  |  :  4  ?   • 

6.  What  part  of  llf  is  5,  {\^%)  and  what  is  11|  :  5  ? 

7.  What  part  of  \  is  f ,  (IS^^)  and  what  is  |-  :  -^  ? 

8.  What  part  of  12i  is  6i,  (ISS,)  and  what  is  12^  :  6J  ? 

9.  What  part  of  .1875  is  .125,  (180.)  and  what  is  .1875 

:.125? 

10.  What  part  of  6.25  is  .625,  (188.)  and  what  is  6.25 
:  .625  ? 

11.  Wliat  part  of  1.16  is  .83,  (184,)  and  what  is  1.16  : 
.83? 

12.  What  part  of  12  hours  is  5h.  15  m.,  (228.)  and  what 
is  12 h.  :  5h.  15m.? 

13.  What  part  of  1£  10s.  is  13s.  6d.,  (230.)  and  what 
is  1£  10s.  :  13s.  6d.  ? 

14.  What  part  of  16s.  6d.  is  $2.50,  (254.)  and  what  is 
16s.  6d.  :  $2.50  ? 


PROPORTION.  171 

S{66«   Model  of  a  Recitation. 

1.    Multiply  25  by  the  ratio  of  7  to  3. 

The  ratio  of  7  to  3  being  ^,  to 


2^==J^^=10f 


multiply  25  by  the  ratio  of  7  to  3  is 


the  same  as  to  multiply  it  by  ^,  that 
is,  (I445)  to  take  ^  of  25,  making 
10^,  which  is  the  answer  required. 

2.  Multiply  8/^  by  63  :  40. 

8yV  =  -VV-'  and  63  :  40 

}^dt       ^iTi  ==  If  J    therefore,    multiply 

I3g  X  40  __  Z^  __  5  5  the    denominator   by  63,  to 

/0   X  $ii         14  ^^         obtain  ^,  {II45)  and  mul- 

2  7  tiply  the  numerator  by  40, 

to  obtain  f  §.  But  both  terms 
cf  this  fraction  having  the  common  factors  9  and  8,  reduce 
tie  fraction  to  its  lowest  terms,  (121j)  before  performing 
tie  operations  indicated  by  the  signs,  (2i53.) 

3.  If  a  man  travel  30  miles   in  7  hours,  what  distance 
\/ould  he  travel  in  12  hours  ? 

If  in  7  hours  he  travel 

2oxj^^^6(i  =  513  jniles  ^^    "^^^^^'  ^^    ^   *^^^^    ^® 

'  ^  ^  *         would  travel  ^  of  30  miles, 

or  ^  miles,  (845)  and  in 

12  hours  he  would  travel  12  times  as  far,  or  s^^^^  =  ^^ 

r=  51^  miles. 

A  shorter  explanation.     12  hours  being  J/-  of  7  hours,  he 

would  travel  in  12  hours  ^^  of  the  distance  that  he  would  ii. 

7  hours.     ^  of  30   miles  is   3qXi8  —  aso  _ 513  j^jies, 

(148.) 

4.  If  5  tons  of  hay  keep  60  sheep  through  the  winter,  how 
much  would  keep  75  sheep  the  same  time  ? 

75  sheep  being  |^  of  60  sheep, 
-^  =  ^  ==  6|  tons.         they  would  require  |4»  or  J,  as 
much  hay.     |-  of  5  tons  is  -^^ 
=  ^5- =  6 J  tons. 

S67.    Exercises  in  multiplying  by  Ratios. 

In  like  manner^  solve  and  explain  the  following  prolilems. 

1.  If  a  piece  of  linen  cost  $24,  what  would  J  of  a  piece 
cost  ? 

2.  If  3  chaldrons  of  coal  cost  $36,  what  part  of  $36,  and 
how  much,  would  1  chaldron  cost  ? 


172  ARITHMETIC. 

3.  At  $4.20  per  box  of  lemons,  what  part  of  $4.20,  and 
how  much,  would  |  of  a  box  cost  ? 

4.  At  $7.50  per  cord,  what  part  of  $7.50,  and  how  much, 
would  I  of  a  cord  of  wood  cost  ? 

5.  At  $.75  per  bushel,  what  part  of  $.75,  and  how  much, 
would  4|-  bushels  of  corn  cost  ? 

6.  If  6  horses  eat  18  bushels  of  oats  in  a  week,  what  part 
of  18  bushels,  and  how  much,  would  5  horses  eat  ? 

7.  If  25  lbs.  of  sugar  cost  $2.25,  what  would  be  the  cost 
of  60  lbs.  ? 

8.  If  5  tons  of  hay  cost  $87.50,  what  part  of  $87.50,  and 
how  much,  would  12  tons  cost  ? 

9.  At  $54  for  9  barrels  of  flour,  what  part  of  9  barrels,  and 
how  much,  could  be  purchased  for  $186  ? 

10.  If  a  vessel  sail  480  miles  in  5  days,  how  long  would 
it  take  her  to  sail  3000  miles  ? 

11.  If  30  cords  of  wood  cost  $200,  what  part  of  $200,  and 
how  much,  would  75  cords  cost  ? 

12.  If  3  books  cost  f  of  a  dollar,  what  part  of  J  of  a  dollar 
and  how  much,  would  8  books  cost  ? 

S^8*  Reduction  of  Complex  Fkactions. 

A  complex  fraction  is  a  fraction  in  which  either  term,  or 
both  terms  are  fractions,  or  mixed  numbers.  It  may  be 
reduced  to  a  simple  fraction  by  multiplying  both  terms  (131) 

284       ~%^ 
by  the  denominators  of  the  terms.     Thus,  ~I,  or  =,  are 

complex  fractions,  and  by  multiplying  both  terms  by  5  and  8, 
or  40,  we  have  V^^'-  =  VA^- 

If  the  denominators  of  the  terms  of  a  complex  fraction  have 
a  common  multiple  (131)  less  than  their  product,  multiply 

I 
both  terms  by  that  least  common  multiple.      Thus,  in  =,  by 

"^  . 
multiplying  both  terms  by  24,  we  have  the  simple  fractioi? 

S69.   Model  of  a  Recitation. 

16.  If  SJlbs.  of  butter  cost  $1-^,  what  would  be  the  price 
of  28|lbs.  at  that  rate  ? 


PROPORTION.  173 

^  yw  Since  $l^,or$|^ 

y^ v^cvx  ^A^         c,A  ^s  ^^^  P^ice  of  8i  lbs. 

rt     ./-^    =  $-=  $4.80.         «^  -V-  lbs.,  divide  it  by 
""XXX^^X^  ""d     ■  65  for  the  price  of  Jib. 

$  and  multiply  that  quo- 

tient, $3-^-5^-5-,  by  8 
f )r  the  price  of  fib.  or  1  lb.  (156)  I  then,  since  the  price  of 
S8f  lb.  or  ^^  lb.  is  required,  divide  the  price  of  1  lb.  SyVxVb* 
ly  5  for  the  price  of  ^  lb.  and  multiply  that  quotient, 
^'tt^fV'x5'  by  143  for  the  price  of  -if^  lb.  or  28f  lb.  (I495) 
naking  $-Y^|^*f==  $2^4  =  $4.80,  which  is  the  answer 
r  squired. 

In  reducing  this  fraction,  -r^-i^^f-i  to  its  lowest  terms, 
(131)  we  divide  both  terms  by  11  by  canceling  the  11  in  the 
cenominator  by  the  11  which  is  a  factor  of  143  in  the  nume- 
rator ;  13,  the  other  factor  of  143,  we  cancel  by  the  13  which 
is  a  factor  of  65  in  the  denominator;  and  5,  the  other  factor 
cf  65,  we  cancel  by.  the  5  which  is  a  factor  of  15  in  the 
r  umerator,  giving  -2^,  or  ^. 

Note.  In  canceling  equal  factors,  there  will  be  less  liability 
t3  mistake,  and  greater  facility  in  reviewing  the  process,  if 
one  continued  line  be  drawn  through  the  two  numbers  con- 
taining the  factor  to  be  canceled. 

A  shorter  explanation.  If  Sl^  or,  S^-,  the  price  of  8J  lb. 
or  V  lb.  be  divided  by  -V-,  (156,)  the  quotient,  »TV>f^'  will 
be  the  price  of  1  lb. ;  and  if  this  price  of  1  lb.  be  multiplied 
by  283-,  or  -i^,  the  number  of  pounds  whose  price  is 
required,  (149,)  the  product  $\^-^^V¥' =  HS  =  $4.80, 
must  be  the  answer  required. 

285.       ^^ 

Or  shorter  still.    28f  lb.  being  -^f ,  or  =^  of  8|  lb.  or  ^  lb. 

^      ¥• 

JA3  143 

would  cost  =:  of  the  price  of  -^^  lb.    ==:  of  $l^y,  or  $4t'  ^^ 

^\'^i^kh  =  $¥'  =  $4.80,  as  before. 

370*    Observation. 

Observe,  that  the  process^  by  either  explanation,  (Q69^) 

is  the  same,  and  consists  of  multiplying^  the  given  price  of  a 

given  quantity  by  the  ratio  of  the  given  quantity  to  the 

REQUIRED  quantity,  or  by  the  part  of  the  given  quantity  that 

15=^ 


174  ARITHMETIC. 

the  required  quantity  must  he.  Labor  often  may  he  saved  by 
mentally  reducing  the  ratio  to  simpler  terms ^  (121 5)  hefore 
writing  it, 

271,  Exercises  in  Multiplying  by  Complex  Ratios. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  If  37  yards  of  broad  cloth  cost  $185,  what  would  5f 
yards  cost  ? 

2.  If  If  of  a  cask  of  wine  cost  $12.50,  what  would  6  such 
casks  cost  ? 

3.  At  $3f  for  4f  yards  of  satinet,  what  would  be  the  cost 
of  25  yards  ? 

4.  If  12  days'  work  cost  $16|,  what  would  5f  days'  work 
cost  ? 

5.  If  1^  of  a  bushel  of  corn  cost  $|-,  what  is  that  a  bushel  ? 

6.  If  IJ  barrels  of  flour  serve  a  family  1^  weeks,  how 
long  would  7^  barrels  serve  them  ? 

7.  If  a  (Company  of  workmen  mow  72^  acres  in  12 1  days, 
how  many  acres  would  they  mow,  at  the  same  rate,  in  Sf 
days  ? 

8.  If  3  yds.  3  qrs.  of  cassimere  cost  $10,  what  would  5 
yards  cost  ? 

9.  If  18  gals.  3  qts.  of  wine  cost  $33.75,  what  would 
43  gals.  3  qts.  cost  ? 

10.  If  2  roods,  25  rods  of  land  cost  $42,  what  would 
5  acres,  3  roods  cost,  at  that  rate  ? 

11.  If  £^jj  sterling  money  make  $2,  how  much  sterling 
money  is  equal  to  $12J  ? 

272.  Illustration  of  the  Inverse  Ratio. 

If  5  men  could  build  a  wall  in  32  days,  in  how  long  time 
could  9  men  build  it  ? 

It  would  take  1  man  5  times  as  long  as  it  would  5  men,  or 
5  times  32  days  ;  but  it  would  take  9  men  only  J  as  long  as 
it  would  1  man,,  or  ^  of  5  times  32  days,  which  is  |  of  32 
days,  or  ^^l^=:i|.o  =  17 J-  days,  the  answer  required. 

S73.    Observation. 

Observe,  (272^)  that,  though  9  men  arc  %  of  5  men,  it 
would  not  take  them  f  of  the  time  that  it  would  take  5  men 
to  build  the  wall,  but  rather ^  |-  of  that  time ;   but  f  w  f 


PROPORTION.  176 

i?JVERTED.  Hence,  f  in  this  example,  and  the  ratio  corre- 
svonding  to  it  in  similar  cases,  being  called  a  direct  ratio, 
f  and  the  ratio  corresponding  to  it  in  similar  cases,  is  called 

en   INVERSE    RATIO. 

tl74:m   Model  of  a  Recitation. 

If  12  cows  consume  a  quantity  of  hay  in  90  days,  how 
inany  cows  would  consume  the  same  hay  in  30  days  ? 

To  consume  the  same  hay  in  30  days  would  require  f§,  or 
(131)  3  times  as  many  cows  as  would  consume  it  in  90 
(lays.     3  times  12  cows  are  36  cows,  the  answer  required. 

^27^.   Exercises  in  Multiplying  by  Inverse  Ratios. 

In  like  manner,  solve  and  explain  the  following  problems. 

1.  If  9  horses  consume  a  ton  of  hay  in  32  days,  how  long 
.vould  it  take  12  horses  to  consume  the  same  hay  ? 

2.  If  72  men  could  do  a  job  of  work  in  15  months,  how 
inany  men,  working*at  the  same  rate,  would  do  the  same  job 
]  n  2  years  ? 

3.  If  a  barrel  of  flour  last  a  family  of  6  persons  12  weeks, 
Jiow  long  would  a  barrel  last  them  if  the  family  be  increased 
i;o  8  persons  ? 

4.  If  a  pail  holding  10  quarts  be  emptied  200  times  to  fill 
a  cistern,  how  much  would  that  vessel  hold  which  must  be 
emptied  75  times  to  fill  the  same  cistern  ? 

5.  4s.  6d.  sterling  money  being  equal  to  5s.  Canada  cur- 
rency, how  much  sterling  money  would  cancel  a  debt  of  £18 
in  Quebec  ? 

6.  How  much  Canada  currency  would  cancel  a  debt  of 
£36  in  London  ? 

7.  5s.  Canada  currency  being  equal  to  6s.  N.  E.  currency, 
how  much  Canada  currency  is  equal  to  £45  N.  E.  currency? 

8.  How  much  sterling  money  is  equal  to  £36  N.  E. 
currency  ? 

9.  6s.  N.  E.  currency  being  equal  to  8s.  N.  Y.  currency, 
how  many  N.  E.  pounds  are  equal  to  72  N.  Y.  pounds  ? 

10.  A  piece  of  land  8  rods  wide  and  20  rods  long  is  an 
acre ;  then  how  long  must  that  acre  be  which  is  12  rods  wide  ? 

11.  A  board  9  in.  wide  and  16  in^long  being  a  square 
foot,  how  wide  must  that  board  be  which  contains  1  sq.  foot, 
and  is  16  feet  long  ? 


it  * 


176 


ARITHMETIC. 


12.  If  a  Stick  of  timber,  the  end  of  which  contains  216  sq. 
inches,  must  be  37J  feet  long  to  be  a  ton,  how  long  must  a 
stick  be  to  measure  a  ton,  the  end  of  which  contains  288  sq. 
inches  ? 

13.  If  the  contents  of  a  cylindrical  tube,  which  measures 
18  inches  in  length  and  144  sq.  inches  on  one  end,  be  emptied 
into  another  tube  the  end  of  which  should  measure  16  sq. 
inches,  how  high  would  the  water  rise  ? 

14.  How  many  yards  of  cloth  |  of  a  yard  wide  would  be 
equal  to  12|  yards  1|  yards  wide  ? 

15.  How  many  yards  of  cloth,  1|  yds.  wide,  would  be 
equal  to  91^  yds.  f  of  a  yard  wide  ? 

16.  What  quantity  of  wheat,  at  $1.25  a  bushel,  should  be 
given  for  10  barrels  of  flour,  at  $5.25  a  barrel  ? 

17.  What  quantity  of  sugar,  at  SJcts.  a  pound,  would  pay 
for  board  12  weeks,  at  $2.75  a  week  ? 

18.  In  how  many  weeks,  at  SIOOO  per  annum,  could 
a  man  earn  as  much  as  another  man  could  in  13  weeks,  at 
$700  per  annum  ? 

376*    Illustration  of  the  Principles  of  Proportion. 

A  large  and  a  sihall  map  of  the  U.  S.  in  order  to  be  correct 
representations  of  the  country,  must  be  of  the  same  shape  ; 
and  the  states,  mountains,  lakes,  rivers,  cities,  towns,  &;c.j 
must  have  the  same  relative  distaiices  on  each  map  ;  that  is, 
all  the  distances  on  each  map,  must  be  in  'proportion  to  the 
corresponding  distances  on  the  other  map.  Thus ;  if  New 
York  is  \  as  far  from  Washington  as  Boston  is  on  one  map, 
it  must  also  be  \  as  far  from  W.  as  B.  is  on  the  other  map. 

If  then,  on  the  larger  map,  the  two  distances  of  B.  and 
N.  Y.  from  W.  be  12  inches,  and  \  of  12  iiiches,  or  6  inches, 
and  on  the  smaller  map,  the  distance  of  B.  from  W.  be  4  in., 
the  distance  of  N.  Y.  from  W.  must  he  \  of  ^  inches,  or  2  in. 
That  is,  the  ratio  (264)  of  the  two  distances  of  B.  and  N.  Y. 
from  W.  on  one  map,  must  be  equal  to  the  ratio  of  the  cor- 
responding distances  on  the  other  map.  Thus,  12  :  6  ==  4 
:  2.  This  expression  constitutes  what  is  called  a  pro- 
vortion. 

Observe,  then,  that  a  proportion  is  composed  of  two  equal 
ratios,  and  that  a  ratio  is  the  relation  of  two  quantities  of  the 
same  kind,  in  regard  to  what  part  (87)  of  the  first,  the 
second  is,  or  how  many  times  the  first  is  contained  by  the 


PROPORTION.  177 

ee'.ond  Thus,  in  the  proportion  12  6  =  4:2,  the  first 
raio  12  :  6,  or  -fir,  is  equal  to  the  second  ratio  4  :  2,  or  |, 
since  each  is  equal  to  ^.  This  proportion  is  read:  The 
ra  Jo  of  12  to  6  equals  the  ratio  of  4  to  2 ;  or  12  is  to  6  as  4 
is  to  2 ;  or,  as  12  is  to  6  so  is  4  to  2. 

The  four  quantities  forming  a  proportion  are  called  pro- 
portionals, or,  the  terms  of  the  pi'oportion. 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
ex '^.rentes,  and  the  second  and  third,  the  means. 

Also,  the  first  terms,  or  divisors  in  ratios,  are  called  the 
atitecedentSj  and  the  second  terms,  or  dividends,  are  called  the 
CO  tsequents. 

Two  equal  fractions  may  become  a  proportion,  by  placing 

th3  denominators  for  antecedents,  and  the   numerators   for 

consequents.     And,  any  four  numbers,  arranged  like  propor- 

tiimals,  form  a  correct  proportion,  if  the  product  of  the  means 

he  equal  to  the  product  of  the  extremes,  since  these  products, 

in  a  correct  proportion,  will  always  be  equal ;  for,  in  a-  correct 

proportion,  the  two  ratios,  or   fractions 

12  :  6==4  :  2         being  equal,  if  they  be  reduced  to  a  com- 

-5-^2- =  f  mon  denominator,  (  139,)  by  multiply- 

y6^^=  |gJ-|        ing  both  terms  of  each  by  the  denomina- 

||  =  ||  tor  of  the  other,  the  numerators  will  he 

equal  also.     But  one  of  these  numerators 

is  the  product  of  the  means,  and  the  other  is  the  product  of 

the  extremes. 

This  truth  is  of  great  practical  utility  in  the  solution  of 
problems  which  involve  proportion ;  since,  by  its  application, 
any  three  terms  of  a  proportion  are  sufficient  for  ascertaining 
the  remaining  term.  For,  if  the  term  wanting  be  an  extreme, 
it  may  be  ascertained  by  dividing  the  product  of  the  means, 
(which  is  also  the  product  of  the  extremes,  one  of  which 
is  known,)  by  the  known  extreme,  (117) ;  or,  if  the  term 
wanting  be  a  mean,  it  may  be  ascertained  by  dividing  the 
product  of  the  extremes,  (which  is  also  the  product  of  the 
means,  one  of  which  is  known,)  by  the  known  mean.  Thus, 
in  the  proportion,  12  :  6==  4  :  2,  the  product  of  the  means 
divided  by  the^^r^^  extreme,  is  ^-^^  =  2,  the  second  extreme ; 
or  divided  by  the  second  extreme,  is  ^^=  12,  the  first 
extreme  ;  and  the  product  of  the  extremes^  divided  by  the  first 
mean,  is  ^^^=4,  the  second  mean;  or,  divided  by  the 
second  mean,  is  ^^^  =  6,  the  first  mean. 


